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Everyone struggles with homework sometimes, but if getting your homework done has become a chronic issue for you, then you may need a little extra help. That’s why we’ve written this article all about how to do homework. Once you’re finished reading it, you’ll know how to do homework (and have tons of new ways to motivate yourself to do homework)!

We’ve broken this article down into a few major sections. You’ll find:

• A diagnostic test to help you figure out why you’re struggling with homework
• A discussion of the four major homework problems students face, along with expert tips for addressing them
• A bonus section with tips for how to do homework fast

By the end of this article, you’ll be prepared to tackle whatever homework assignments your teachers throw at you .

So let’s get started!

How to Do Homework: Figure Out Your Struggles 

Sometimes it feels like everything is standing between you and getting your homework done. But the truth is, most people only have one or two major roadblocks that are keeping them from getting their homework done well and on time. 

The best way to figure out how to get motivated to do homework starts with pinpointing the issues that are affecting your ability to get your assignments done. That’s why we’ve developed a short quiz to help you identify the areas where you’re struggling. 

Take the quiz below and record your answers on your phone or on a scrap piece of paper. Keep in mind there are no wrong answers! 

1. You’ve just been assigned an essay in your English class that’s due at the end of the week. What’s the first thing you do?

A. Keep it in mind, even though you won’t start it until the day before it’s due  B. Open up your planner. You’ve got to figure out when you’ll write your paper since you have band practice, a speech tournament, and your little sister’s dance recital this week, too.  C. Groan out loud. Another essay? You could barely get yourself to write the last one!  D. Start thinking about your essay topic, which makes you think about your art project that’s due the same day, which reminds you that your favorite artist might have just posted to Instagram...so you better check your feed right now. 

2. Your mom asked you to pick up your room before she gets home from work. You’ve just gotten home from school. You decide you’ll tackle your chores: 

A. Five minutes before your mom walks through the front door. As long as it gets done, who cares when you start?  B. As soon as you get home from your shift at the local grocery store.  C. After you give yourself a 15-minute pep talk about how you need to get to work.  D. You won’t get it done. Between texts from your friends, trying to watch your favorite Netflix show, and playing with your dog, you just lost track of time! 

3. You’ve signed up to wash dogs at the Humane Society to help earn money for your senior class trip. You: 

A. Show up ten minutes late. You put off leaving your house until the last minute, then got stuck in unexpected traffic on the way to the shelter.  B. Have to call and cancel at the last minute. You forgot you’d already agreed to babysit your cousin and bake cupcakes for tomorrow’s bake sale.  C. Actually arrive fifteen minutes early with extra brushes and bandanas you picked up at the store. You’re passionate about animals, so you’re excited to help out! D. Show up on time, but only get three dogs washed. You couldn’t help it: you just kept getting distracted by how cute they were!

4. You have an hour of downtime, so you decide you’re going to watch an episode of The Great British Baking Show. You: 

A. Scroll through your social media feeds for twenty minutes before hitting play, which means you’re not able to finish the whole episode. Ugh! You really wanted to see who was sent home!  B. Watch fifteen minutes until you remember you’re supposed to pick up your sister from band practice before heading to your part-time job. No GBBO for you!  C. You finish one episode, then decide to watch another even though you’ve got SAT studying to do. It’s just more fun to watch people make scones.  D. Start the episode, but only catch bits and pieces of it because you’re reading Twitter, cleaning out your backpack, and eating a snack at the same time.

5. Your teacher asks you to stay after class because you’ve missed turning in two homework assignments in a row. When she asks you what’s wrong, you say: 

A. You planned to do your assignments during lunch, but you ran out of time. You decided it would be better to turn in nothing at all than submit unfinished work.  B. You really wanted to get the assignments done, but between your extracurriculars, family commitments, and your part-time job, your homework fell through the cracks.  C. You have a hard time psyching yourself to tackle the assignments. You just can’t seem to find the motivation to work on them once you get home.  D. You tried to do them, but you had a hard time focusing. By the time you realized you hadn’t gotten anything done, it was already time to turn them in. 

Like we said earlier, there are no right or wrong answers to this quiz (though your results will be better if you answered as honestly as possible). Here’s how your answers break down: 

• If your answers were mostly As, then your biggest struggle with doing homework is procrastination. 
• If your answers were mostly Bs, then your biggest struggle with doing homework is time management. 
• If your answers were mostly Cs, then your biggest struggle with doing homework is motivation. 
• If your answers were mostly Ds, then your biggest struggle with doing homework is getting distracted. 

Now that you’ve identified why you’re having a hard time getting your homework done, we can help you figure out how to fix it! Scroll down to find your core problem area to learn more about how you can start to address it. 

And one more thing: you’re really struggling with homework, it’s a good idea to read through every section below. You may find some additional tips that will help make homework less intimidating. 

How to Do Homework When You’re a Procrastinator  

Merriam Webster defines “procrastinate” as “to put off intentionally and habitually.” In other words, procrastination is when you choose to do something at the last minute on a regular basis. If you’ve ever found yourself pulling an all-nighter, trying to finish an assignment between periods, or sprinting to turn in a paper minutes before a deadline, you’ve experienced the effects of procrastination. 

If you’re a chronic procrastinator, you’re in good company. In fact, one study found that 70% to 95% of undergraduate students procrastinate when it comes to doing their homework. Unfortunately, procrastination can negatively impact your grades. Researchers have found that procrastination can lower your grade on an assignment by as much as five points ...which might not sound serious until you realize that can mean the difference between a B- and a C+. 

Procrastination can also negatively affect your health by increasing your stress levels , which can lead to other health conditions like insomnia, a weakened immune system, and even heart conditions. Getting a handle on procrastination can not only improve your grades, it can make you feel better, too! 

The big thing to understand about procrastination is that it’s not the result of laziness. Laziness is defined as being “disinclined to activity or exertion.” In other words, being lazy is all about doing nothing. But a s this Psychology Today article explains , procrastinators don’t put things off because they don’t want to work. Instead, procrastinators tend to postpone tasks they don’t want to do in favor of tasks that they perceive as either more important or more fun. Put another way, procrastinators want to do things...as long as it’s not their homework! 

3 Tips f or Conquering Procrastination 

Because putting off doing homework is a common problem, there are lots of good tactics for addressing procrastination. Keep reading for our three expert tips that will get your homework habits back on track in no time. 

#1: Create a Reward System

Like we mentioned earlier, procrastination happens when you prioritize other activities over getting your homework done. Many times, this happens because homework...well, just isn’t enjoyable. But you can add some fun back into the process by rewarding yourself for getting your work done. 

Here’s what we mean: let’s say you decide that every time you get your homework done before the day it’s due, you’ll give yourself a point. For every five points you earn, you’ll treat yourself to your favorite dessert: a chocolate cupcake! Now you have an extra (delicious!) incentive to motivate you to leave procrastination in the dust. 

If you’re not into cupcakes, don’t worry. Your reward can be anything that motivates you . Maybe it’s hanging out with your best friend or an extra ten minutes of video game time. As long as you’re choosing something that makes homework worth doing, you’ll be successful. 

#2: Have a Homework Accountability Partner 

If you’re having trouble getting yourself to start your homework ahead of time, it may be a good idea to call in reinforcements . Find a friend or classmate you can trust and explain to them that you’re trying to change your homework habits. Ask them if they’d be willing to text you to make sure you’re doing your homework and check in with you once a week to see if you’re meeting your anti-procrastination goals. 

Sharing your goals can make them feel more real, and an accountability partner can help hold you responsible for your decisions. For example, let’s say you’re tempted to put off your science lab write-up until the morning before it’s due. But you know that your accountability partner is going to text you about it tomorrow...and you don’t want to fess up that you haven’t started your assignment. A homework accountability partner can give you the extra support and incentive you need to keep your homework habits on track. 

#3: Create Your Own Due Dates 

If you’re a life-long procrastinator, you might find that changing the habit is harder than you expected. In that case, you might try using procrastination to your advantage! If you just can’t seem to stop doing your work at the last minute, try setting your own due dates for assignments that range from a day to a week before the assignment is actually due. 

Here’s what we mean. Let’s say you have a math worksheet that’s been assigned on Tuesday and is due on Friday. In your planner, you can write down the due date as Thursday instead. You may still put off your homework assignment until the last minute...but in this case, the “last minute” is a day before the assignment’s real due date . This little hack can trick your procrastination-addicted brain into planning ahead! 

If you feel like Kevin Hart in this meme, then our tips for doing homework when you're busy are for you. 

How to Do Homework When You’re too Busy

If you’re aiming to go to a top-tier college , you’re going to have a full plate. Because college admissions is getting more competitive, it’s important that you’re maintaining your grades , studying hard for your standardized tests , and participating in extracurriculars so your application stands out. A packed schedule can get even more hectic once you add family obligations or a part-time job to the mix. 

If you feel like you’re being pulled in a million directions at once, you’re not alone. Recent research has found that stress—and more severe stress-related conditions like anxiety and depression— are a major problem for high school students . In fact, one study from the American Psychological Association found that during the school year, students’ stress levels are higher than those of the adults around them. 

For students, homework is a major contributor to their overall stress levels . Many high schoolers have multiple hours of homework every night , and figuring out how to fit it into an already-packed schedule can seem impossible. 

3 Tips for Fitting Homework Into Your Busy Schedule

While it might feel like you have literally no time left in your schedule, there are still ways to make sure you’re able to get your homework done and meet your other commitments. Here are our expert homework tips for even the busiest of students. 

#1: Make a Prioritized To-Do List 

You probably already have a to-do list to keep yourself on track. The next step is to prioritize the items on your to-do list so you can see what items need your attention right away. 

Here’s how it works: at the beginning of each day, sit down and make a list of all the items you need to get done before you go to bed. This includes your homework, but it should also take into account any practices, chores, events, or job shifts you may have. Once you get everything listed out, it’s time to prioritize them using the labels A, B, and C. Here’s what those labels mean:

• A Tasks : tasks that have to get done—like showing up at work or turning in an assignment—get an A. 
• B Tasks : these are tasks that you would like to get done by the end of the day but aren’t as time sensitive. For example, studying for a test you have next week could be a B-level task. It’s still important, but it doesn’t have to be done right away.
• C Tasks: these are tasks that aren’t very important and/or have no real consequences if you don’t get them done immediately. For instance, if you’re hoping to clean out your closet but it’s not an assigned chore from your parents, you could label that to-do item with a C.

Prioritizing your to-do list helps you visualize which items need your immediate attention, and which items you can leave for later. A prioritized to-do list ensures that you’re spending your time efficiently and effectively, which helps you make room in your schedule for homework. So even though you might really want to start making decorations for Homecoming (a B task), you’ll know that finishing your reading log (an A task) is more important. 

#2: Use a Planner With Time Labels

Your planner is probably packed with notes, events, and assignments already. (And if you’re not using a planner, it’s time to start!) But planners can do more for you than just remind you when an assignment is due. If you’re using a planner with time labels, it can help you visualize how you need to spend your day.

A planner with time labels breaks your day down into chunks, and you assign tasks to each chunk of time. For example, you can make a note of your class schedule with assignments, block out time to study, and make sure you know when you need to be at practice. Once you know which tasks take priority, you can add them to any empty spaces in your day. 

Planning out how you spend your time not only helps you use it wisely, it can help you feel less overwhelmed, too . We’re big fans of planners that include a task list ( like this one ) or have room for notes ( like this one ). 

#3: Set Reminders on Your Phone 

If you need a little extra nudge to make sure you’re getting your homework done on time, it’s a good idea to set some reminders on your phone. You don’t need a fancy app, either. You can use your alarm app to have it go off at specific times throughout the day to remind you to do your homework. This works especially well if you have a set homework time scheduled. So if you’ve decided you’re doing homework at 6:00 pm, you can set an alarm to remind you to bust out your books and get to work. 

If you use your phone as your planner, you may have the option to add alerts, emails, or notifications to scheduled events . Many calendar apps, including the one that comes with your phone, have built-in reminders that you can customize to meet your needs. So if you block off time to do your homework from 4:30 to 6:00 pm, you can set a reminder that will pop up on your phone when it’s time to get started. 

This dog isn't judging your lack of motivation...but your teacher might. Keep reading for tips to help you motivate yourself to do your homework.

How to Do Homework When You’re Unmotivated 

At first glance, it may seem like procrastination and being unmotivated are the same thing. After all, both of these issues usually result in you putting off your homework until the very last minute. 

But there’s one key difference: many procrastinators are working, they’re just prioritizing work differently. They know they’re going to start their homework...they’re just going to do it later. 

Conversely, people who are unmotivated to do homework just can’t find the willpower to tackle their assignments. Procrastinators know they’ll at least attempt the homework at the last minute, whereas people who are unmotivated struggle with convincing themselves to do it at a ll. For procrastinators, the stress comes from the inevitable time crunch. For unmotivated people, the stress comes from trying to convince themselves to do something they don’t want to do in the first place. 

Here are some common reasons students are unmotivated in doing homework : 

• Assignments are too easy, too hard, or seemingly pointless 
• Students aren’t interested in (or passionate about) the subject matter
• Students are intimidated by the work and/or feels like they don’t understand the assignment 
• Homework isn’t fun, and students would rather spend their time on things that they enjoy 

To sum it up: people who lack motivation to do their homework are more likely to not do it at all, or to spend more time worrying about doing their homework than...well, actually doing it.

3 Tips for How to Get Motivated to Do Homework

The key to getting homework done when you’re unmotivated is to figure out what does motivate you, then apply those things to homework. It sounds tricky...but it’s pretty simple once you get the hang of it! Here are our three expert tips for motivating yourself to do your homework. 

#1: Use Incremental Incentives

When you’re not motivated, it’s important to give yourself small rewards to stay focused on finishing the task at hand. The trick is to keep the incentives small and to reward yourself often. For example, maybe you’re reading a good book in your free time. For every ten minutes you spend on your homework, you get to read five pages of your book. Like we mentioned earlier, make sure you’re choosing a reward that works for you! 

So why does this technique work? Using small rewards more often allows you to experience small wins for getting your work done. Every time you make it to one of your tiny reward points, you get to celebrate your success, which gives your brain a boost of dopamine . Dopamine helps you stay motivated and also creates a feeling of satisfaction when you complete your homework !  

#2: Form a Homework Group 

If you’re having trouble motivating yourself, it’s okay to turn to others for support. Creating a homework group can help with this. Bring together a group of your friends or classmates, and pick one time a week where you meet and work on homework together. You don’t have to be in the same class, or even taking the same subjects— the goal is to encourage one another to start (and finish!) your assignments. 

Another added benefit of a homework group is that you can help one another if you’re struggling to understand the material covered in your classes. This is especially helpful if your lack of motivation comes from being intimidated by your assignments. Asking your friends for help may feel less scary than talking to your teacher...and once you get a handle on the material, your homework may become less frightening, too. 

#3: Change Up Your Environment 

If you find that you’re totally unmotivated, it may help if you find a new place to do your homework. For example, if you’ve been struggling to get your homework done at home, try spending an extra hour in the library after school instead. The change of scenery can limit your distractions and give you the energy you need to get your work done. 

If you’re stuck doing homework at home, you can still use this tip. For instance, maybe you’ve always done your homework sitting on your bed. Try relocating somewhere else, like your kitchen table, for a few weeks. You may find that setting up a new “homework spot” in your house gives you a motivational lift and helps you get your work done. 

Social media can be a huge problem when it comes to doing homework. We have advice for helping you unplug and regain focus.

How to Do Homework When You’re Easily Distracted

We live in an always-on world, and there are tons of things clamoring for our attention. From friends and family to pop culture and social media, it seems like there’s always something (or someone!) distracting us from the things we need to do.

The 24/7 world we live in has affected our ability to focus on tasks for prolonged periods of time. Research has shown that over the past decade, an average person’s attention span has gone from 12 seconds to eight seconds . And when we do lose focus, i t takes people a long time to get back on task . One study found that it can take as long as 23 minutes to get back to work once we’ve been distracte d. No wonder it can take hours to get your homework done! 

3 Tips to Improve Your Focus

If you have a hard time focusing when you’re doing your homework, it’s a good idea to try and eliminate as many distractions as possible. Here are three expert tips for blocking out the noise so you can focus on getting your homework done. 

#1: Create a Distraction-Free Environment

Pick a place where you’ll do your homework every day, and make it as distraction-free as possible. Try to find a location where there won’t be tons of noise, and limit your access to screens while you’re doing your homework. Put together a focus-oriented playlist (or choose one on your favorite streaming service), and put your headphones on while you work. 

You may find that other people, like your friends and family, are your biggest distraction. If that’s the case, try setting up some homework boundaries. Let them know when you’ll be working on homework every day, and ask them if they’ll help you keep a quiet environment. They’ll be happy to lend a hand! 

#2: Limit Your Access to Technology 

We know, we know...this tip isn’t fun, but it does work. For homework that doesn’t require a computer, like handouts or worksheets, it’s best to put all your technology away . Turn off your television, put your phone and laptop in your backpack, and silence notifications on any wearable tech you may be sporting. If you listen to music while you work, that’s fine...but make sure you have a playlist set up so you’re not shuffling through songs once you get started on your homework. 

If your homework requires your laptop or tablet, it can be harder to limit your access to distractions. But it’s not impossible! T here are apps you can download that will block certain websites while you’re working so that you’re not tempted to scroll through Twitter or check your Facebook feed. Silence notifications and text messages on your computer, and don’t open your email account unless you absolutely have to. And if you don’t need access to the internet to complete your assignments, turn off your WiFi. Cutting out the online chatter is a great way to make sure you’re getting your homework done. 

#3: Set a Timer (the Pomodoro Technique)

Have you ever heard of the Pomodoro technique ? It’s a productivity hack that uses a timer to help you focus!

Here’s how it works: first, set a timer for 25 minutes. This is going to be your work time. During this 25 minutes, all you can do is work on whatever homework assignment you have in front of you. No email, no text messaging, no phone calls—just homework. When that timer goes off, you get to take a 5 minute break. Every time you go through one of these cycles, it’s called a “pomodoro.” For every four pomodoros you complete, you can take a longer break of 15 to 30 minutes.

The pomodoro technique works through a combination of boundary setting and rewards. First, it gives you a finite amount of time to focus, so you know that you only have to work really hard for 25 minutes. Once you’ve done that, you’re rewarded with a short break where you can do whatever you want. Additionally, tracking how many pomodoros you complete can help you see how long you’re really working on your homework. (Once you start using our focus tips, you may find it doesn’t take as long as you thought!)

Two Bonus Tips for How to Do Homework Fast

Even if you’re doing everything right, there will be times when you just need to get your homework done as fast as possible. (Why do teachers always have projects due in the same week? The world may never know.)

The problem with speeding through homework is that it’s easy to make mistakes. While turning in an assignment is always better than not submitting anything at all, you want to make sure that you’re not compromising quality for speed. Simply put, the goal is to get your homework done quickly and still make a good grade on the assignment! 

Here are our two bonus tips for getting a decent grade on your homework assignments , even when you’re in a time crunch. 

#1: Do the Easy Parts First 

This is especially true if you’re working on a handout with multiple questions. Before you start working on the assignment, read through all the questions and problems. As you do, make a mark beside the questions you think are “easy” to answer . 

Once you’ve finished going through the whole assignment, you can answer these questions first. Getting the easy questions out of the way as quickly as possible lets you spend more time on the trickier portions of your homework, which will maximize your assignment grade. 

(Quick note: this is also a good strategy to use on timed assignments and tests, like the SAT and the ACT !) 

#2: Pay Attention in Class 

Homework gets a lot easier when you’re actively learning the material. Teachers aren’t giving you homework because they’re mean or trying to ruin your weekend... it’s because they want you to really understand the course material. Homework is designed to reinforce what you’re already learning in class so you’ll be ready to tackle harder concepts later.

When you pay attention in class, ask questions, and take good notes, you’re absorbing the information you’ll need to succeed on your homework assignments. (You’re stuck in class anyway, so you might as well make the most of it!) Not only will paying attention in class make your homework less confusing, it will also help it go much faster, too.

What’s Next?

If you’re looking to improve your productivity beyond homework, a good place to begin is with time management. After all, we only have so much time in a day...so it’s important to get the most out of it! To get you started, check out this list of the 12 best time management techniques that you can start using today.

You may have read this article because homework struggles have been affecting your GPA. Now that you’re on the path to homework success, it’s time to start being proactive about raising your grades. This article teaches you everything you need to know about raising your GPA so you can

Now you know how to get motivated to do homework...but what about your study habits? Studying is just as critical to getting good grades, and ultimately getting into a good college . We can teach you how to study bette r in high school. (We’ve also got tons of resources to help you study for your ACT and SAT exams , too!)

These recommendations are based solely on our knowledge and experience. If you purchase an item through one of our links, PrepScholar may receive a commission.

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3.2: Solve Applications with Systems of Equations

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Learning Objectives

By the end of this section, you will be able to:

• Solve direct translation applications
• Solve geometry applications

Solve uniform motion applications

Before you get started, take this readiness quiz.

• The sum of twice a number and nine is 31. Find the number. If you missed this problem, review [link] .
• Twins Jon and Ron together earned $96,000 last year. Ron earned $8000 more than three times what Jon earned. How much did each of the twins earn? If you missed this problem, review [link] .
• An express train and a local train leave Pittsburgh to travel to Washington, D.C. The express train can make the trip in four hours and the local train takes five hours for the trip. The speed of the express train is 12 miles per hour faster than the speed of the local train. Find the speed of both trains. If you missed this problem, review [link] .

Solve Direct Translation Applications

Systems of linear equations are very useful for solving applications. Some people find setting up word problems with two variables easier than setting them up with just one variable. To solve an application, we’ll first translate the words into a system of linear equations. Then we will decide the most convenient method to use, and then solve the system.

SOLVE APPLICATIONS WITH SYSTEMS OF EQUATIONS.

• Read the problem. Make sure all the words and ideas are understood.
• Identify what we are looking for.
• Name what we are looking for. Choose variables to represent those quantities.
• Translate into a system of equations.
• Solve the system of equations using good algebra techniques.
• Check the answer in the problem and make sure it makes sense.
• Answer the question with a complete sentence.

We solved number problems with one variable earlier. Let’s see how differently it works using two variables.

The sum of two numbers is zero. One number is nine less than the other. Find the numbers.

Example \(\PageIndex{2}\)

The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.

Example \(\PageIndex{3}\)

The sum of two numbers is \(−6\). One number is 10 less than the other. Find the numbers.

\(2, −8\)

Heather has been offered two options for her salary as a trainer at the gym. Option A would pay her $25,000 plus $15 for each training session. Option B would pay her \($10,000+$40\) for each training session. How many training sessions would make the salary options equal?

Example \(\PageIndex{5}\)

Geraldine has been offered positions by two insurance companies. The first company pays a salary of $12,000 plus a commission of $100 for each policy sold. The second pays a salary of $20,000 plus a commission of $50 for each policy sold. How many policies would need to be sold to make the total pay the same?

160 policies

Example \(\PageIndex{6}\)

Kenneth currently sells suits for company A at a salary of $22,000 plus a $10 commission for each suit sold. Company B offers him a position with a salary of $28,000 plus a $4 commission for each suit sold. How many suits would Kenneth need to sell for the options to be equal?

As you solve each application, remember to analyze which method of solving the system of equations would be most convenient.

Example \(\PageIndex{7}\)

Translate to a system of equations and then solve:

When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for 20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes on the elliptical trainer and 30 minutes circuit training she burned 473 calories. How many calories does she burn for each minute on the elliptical trainer? How many calories for each minute of circuit training?

Example \(\PageIndex{8}\)

Mark went to the gym and did 40 minutes of Bikram hot yoga and 10 minutes of jumping jacks. He burned 510 calories. The next time he went to the gym, he did 30 minutes of Bikram hot yoga and 20 minutes of jumping jacks burning 470 calories. How many calories were burned for each minute of yoga? How many calories were burned for each minute of jumping jacks?

Mark burned 11 calories for each minute of yoga and 7 calories for each minute of jumping jacks.

Example \(\PageIndex{9}\)

Erin spent 30 minutes on the rowing machine and 20 minutes lifting weights at the gym and burned 430 calories. During her next visit to the gym she spent 50 minutes on the rowing machine and 10 minutes lifting weights and burned 600 calories. How many calories did she burn for each minutes on the rowing machine? How many calories did she burn for each minute of weight lifting?

Erin burned 11 calories for each minute on the rowing machine and 5 calories for each minute of weight lifting.

Solve Geometry Applications

We will now solve geometry applications using systems of linear equations. We will need to add complementary angles and supplementary angles to our list some properties of angles.

The measures of two complementary angles add to 90 degrees. The measures of two supplementary angles add to 180 degrees.

COMPLEMENTARY AND SUPPLEMENTARY ANGLES

Two angles are complementary if the sum of the measures of their angles is 90 degrees.

Two angles are supplementary if the sum of the measures of their angles is 180 degrees

If two angles are complementary, we say that one angle is the complement of the other.

If two angles are supplementary, we say that one angle is the supplement of the other.

Example \(\PageIndex{10}\)

Translate to a system of equations and then solve.

The difference of two complementary angles is 26 degrees. Find the measures of the angles.

\(\begin{array} {ll} {\textbf{Step 1. Read }\text{the problem.}} &{} \\ {\textbf{Step 2. Identify }\text{what we are looking for.}} &{\text{We are looking for the measure of each}} \\ {} &{\text{angle.}} \\ {\textbf{Step 3. Name }\text{what we are looking for.}} &{\text{Let} x=\text{ the measure of the first angle.}} \\ {} &{\hspace{3mm} y= \text{ the measure of the second angle}} \\ {\textbf{Step 4. Translate }\text{into a system of}} &{\text{The angles are complementary.}} \\ {\text{equations.}} &{\hspace{15mm} x+y=90} \\ {} &{\text{The difference of the two angles is 26}} \\ {} &{\text{degrees.}} \\ {} &{\hspace{15mm} x−y=26} \\ {} &{} \\ {} &{} \\ {\text{The system is shown.}} &{\hspace{15mm} \left\{ \begin{array} {l} x+y=90 \\ x−y=26 \end{array} \right. } \\ {} &{} \\ {} &{} \\ {\textbf{Step 5. Solve }\text{the system of equations} } &{\hspace{15mm} \left\{ \begin{array} {l} x+y=90 \\ \underline{x−y=26} \end{array} \right. } \\ {\text{by elimination.}} &{\hspace{21mm} 2x\hspace{4mm}=116} \\ {} &{\hspace{28mm} x=58} \\ {} &{} \\ {} &{} \\ {\text{Substitute }x=58\text{ into the first equation.}} &{\hspace{15mm} x+y=90} \\ {} &{\hspace{14mm} 58+y=90} \\ {} &{\hspace{22mm} y=32} \\ {\textbf{Step 6. Check }\text{the answer in the problem.}} &{} \\ {} &{} \\ {} &{} \\ {} &{} \\ {\hspace{15mm} 58+32=90\checkmark} &{} \\ {\hspace{15mm} 58−32=26\checkmark} &{} \\ {\textbf{Step 7. Answer }\text{the question.}} &{\text{The angle measures are 58 and 32 degrees.}} \end{array} \)

Example \(\PageIndex{11}\)

The difference of two complementary angles is 20 degrees. Find the measures of the angles.

The angle measures are 55 and 35.

Example \(\PageIndex{12}\)

The difference of two complementary angles is 80 degrees. Find the measures of the angles.

The angle measures are 5 and 85.

In the next example, we remember that the measures of supplementary angles add to 180.

Example \(\PageIndex{13}\)

Two angles are supplementary. The measure of the larger angle is twelve degrees less than five times the measure of the smaller angle. Find the measures of both angles.

Example \(\PageIndex{14}\)

Two angles are supplementary. The measure of the larger angle is 12 degrees more than three times the smaller angle. Find the measures of the angles.

The angle measures are 42 and 138.

Example \(\PageIndex{15}\)

Two angles are supplementary. The measure of the larger angle is 18 less than twice the measure of the smaller angle. Find the measures of the angles.

The angle measures are 66 and 114.

Recall that the angles of a triangle add up to 180 degrees. A right triangle has one angle that is 90 degrees. What does that tell us about the other two angles? In the next example we will be finding the measures of the other two angles.

Example \(\PageIndex{16}\)

The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. Find the measures of both angles.

We will draw and label a figure.

Example \(\PageIndex{17}\)

The measure of one of the small angles of a right triangle is 2 more than 3 times the measure of the other small angle. Find the measure of both angles.

Example \(\PageIndex{18}\)

The measure of one of the small angles of a right triangle is 18 less than twice the measure of the other small angle. Find the measure of both angles.

Often it is helpful when solving geometry applications to draw a picture to visualize the situation.

Example \(\PageIndex{19}\)

Randall has 125 feet of fencing to enclose the part of his backyard adjacent to his house. He will only need to fence around three sides, because the fourth side will be the wall of the house. He wants the length of the fenced yard (parallel to the house wall) to be 5 feet more than four times as long as the width. Find the length and the width.

Example \(\PageIndex{20}\)

Mario wants to put a fence around the pool in his backyard. Since one side is adjacent to the house, he will only need to fence three sides. There are two long sides and the one shorter side is parallel to the house. He needs 155 feet of fencing to enclose the pool. The length of the long side is 10 feet less than twice the width. Find the length and width of the pool area to be enclosed.

The length is 60 feet and the width is 35 feet.

Example \(\PageIndex{21}\)

Alexis wants to build a rectangular dog run in her yard adjacent to her neighbor’s fence. She will use 136 feet of fencing to completely enclose the rectangular dog run. The length of the dog run along the neighbor’s fence will be 16 feet less than twice the width. Find the length and width of the dog run.

The length is 60 feet and the width is 38 feet.

We used a table to organize the information in uniform motion problems when we introduced them earlier. We’ll continue using the table here. The basic equation was \(D=rt\) where D is the distance traveled, r is the rate, and t is the time.

Our first example of a uniform motion application will be for a situation similar to some we have already seen, but now we can use two variables and two equations.

Example \(\PageIndex{22}\)

Joni left St. Louis on the interstate, driving west towards Denver at a speed of 65 miles per hour. Half an hour later, Kelly left St. Louis on the same route as Joni, driving 78 miles per hour. How long will it take Kelly to catch up to Joni?

A diagram is useful in helping us visualize the situation.

Identify and name what we are looking for. A chart will help us organize the data. We know the rates of both Joni and Kelly, and so we enter them in the chart. We are looking for the length of time Kelly, k , and Joni, j , will each drive.

Since \(D=r·t\) we can fill in the Distance column.

To make the system of equations, we must recognize that Kelly and Joni will drive the same distance. So,

\(\hspace{85mm} 65j=78k \nonumber \)

Also, since Kelly left later, her time will be \(\frac{1}{2}\) hour less than Joni’s time. So,

\( \hspace{105mm} k=j-\frac{1}{2} \nonumber \)

\(\begin{array} {ll} {\text{Now we have the system.}} &{\left\{ \begin{array} {l} k=j−\frac{1}{2} \\ 65j=78k \end{array} \right.} \\ {\textbf{Solve }\text{the system of equations by substitution.}} &{} \\ {} &{} \\ {\text{Substitute }k=j−12\text{ into the second equation,}} &{} \\ {\text{then solve for }j.} &{} \\ {} &{65j=78k} \\ {} &{65j=78(j−\frac{1}{2})} \\ {} &{65j=78j−39} \\ {} &{−13j=−39} \\ {} &{j=3} \\{\begin{array} {l} {\text{To find Kelly’s time, substitute }j=3 \text{ into the first}} \\ {\text{equation, then solve for }k.} \end{array} } &{k=j−\frac{1}{2}} \\ {} &{k=3−\frac{1}{2} } \\ {} &{k=\frac{5}{2} \text{ or } k=2\frac{1}{2}} \\ {\textbf{Check }\text{the answer in the problem.}} &{} \\ {\begin{array} {lllll} {\text{Joni}} &{3 \text{ hours}} &{(65\text{ mph})} &= &{195\text{ miles}} \\ {\text{Kelly}} &{2\frac{1}{2} \text{ hours}} &{(78\text{ mph})} &= &{195\text{ miles}} \end{array}} &{} \\ {\text{Yes, they will have traveled the same distance}} &{} \\{\text{when they meet.}} &{} \\ {\textbf{Answer }\text{the question.}} &{} \\ {} &{\text{Kelly will catch up to Joni in}} \\ {} &{2\frac{1}{2}\text{ hours. By then, Joni will}} \\ {} &{\text{have traveled }3 \text{ hours.}} \\ \end{array}\)

Example \(\PageIndex{23}\)

Mitchell left Detroit on the interstate driving south towards Orlando at a speed of 60 miles per hour. Clark left Detroit 1 hour later traveling at a speed of 75 miles per hour, following the same route as Mitchell. How long will it take Clark to catch Mitchell?

It will take Clark 4 hours to catch Mitchell.

Example \(\PageIndex{24}\)

Charlie left his mother’s house traveling at an average speed of 36 miles per hour. His sister Sally left 15 minutes \((\frac{1}{4} \text{ hour})\) later traveling the same route at an average speed of 42 miles per hour. How long before Sally catches up to Charlie?

It will take Sally \(112\) hours to catch up to Charlie.

Many real-world applications of uniform motion arise because of the effects of currents—of water or air—on the actual speed of a vehicle. Cross-country airplane flights in the United States generally take longer going west than going east because of the prevailing wind currents.

Let’s take a look at a boat travelling on a river. Depending on which way the boat is going, the current of the water is either slowing it down or speeding it up.

The images below show how a river current affects the speed at which a boat is actually travelling. We’ll call the speed of the boat in still water b and the speed of the river current c .

The boat is going downstream, in the same direction as the river current. The current helps push the boat, so the boat’s actual speed is faster than its speed in still water. The actual speed at which the boat is moving is \(b+c\).

Now, the boat is going upstream, opposite to the river current. The current is going against the boat, so the boat’s actual speed is slower than its speed in still water. The actual speed of the boat is \(b−c\).

We’ll put some numbers to this situation in the next example.

Example \(\PageIndex{25}\)

A river cruise ship sailed 60 miles downstream for 4 hours and then took 5 hours sailing upstream to return to the dock. Find the speed of the ship in still water and the speed of the river current.

Example \(\PageIndex{26}\)

A Mississippi river boat cruise sailed 120 miles upstream for 12 hours and then took 10 hours to return to the dock. Find the speed of the river boat in still water and the speed of the river current.

The rate of the boat is 11 mph and the rate of the current is 1 mph.

Example \(\PageIndex{27}\)

Jason paddled his canoe 24 miles upstream for 4 hours. It took him 3 hours to paddle back. Find the speed of the canoe in still water and the speed of the river current.

The speed of the canoe is 7 mph and the speed of the current is 1 mph.

Wind currents affect airplane speeds in the same way as water currents affect boat speeds. We’ll see this in the next example. A wind current in the same direction as the plane is flying is called a tailwind . A wind current blowing against the direction of the plane is called a headwind .

Example \(\PageIndex{28}\)

A private jet can fly 1,095 miles in three hours with a tailwind but only 987 miles in three hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

Example \(\PageIndex{29}\)

A small jet can fly 1,325 miles in 5 hours with a tailwind but only 1,035 miles in 5 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

The speed of the jet is 235 mph and the speed of the wind is 30 mph.

Example \(\PageIndex{30}\)

A commercial jet can fly 1,728 miles in 4 hours with a tailwind but only 1,536 miles in 4 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

The speed of the jet is 408 mph and the speed of the wind is 24 mph.

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• Chapter 6 Fraction Equivalence and Comparison
• Chapter 7 Add and Subtract Fractions
• Chapter 8 Multiply Fractions by Whole Numbers
• Chapter 9 Relate Fractions and Decimals
• Chapter 10 Two-Dimensional Figures
• Chapter 11 Angles
• Chapter 12Relative Sizes of Measurement Units
• Chapter 13 Algebra: Perimeter and Area

Grade 4 Homework Practice FL.

Common Core – Grade 4 – Practice Book

• Chapter 1 Place Value, Addition, and Subtraction to One Million  (Pages 1- 20)
• Chapter 2 Multiply by 1-Digit Numbers  (Pages 21 – 47)
• Chapter 3 Multiply 2-Digit Numbers  (Pages 49- 65)
• Chapter 4 Divide by 1-Digit Numbers  (Pages 67 – 93)
• Chapter 5 Factors, Multiples, and Patterns  (Pages 95 – 109)
• Chapter 6 Fraction Equivalence and Comparison  (Pages 111 – 129)
• Chapter 7 Add and Subtract Fractions  (Pages 131 – 153)
• Chapter 8 Multiply Fractions by Whole Numbers  (Pages 155- 167)
• Chapter 9 Relate Fractions and Decimals  (Pages 169- 185)
• Chapter 10 Two-Dimensional Figures  (Pages 187- 204)
• Chapter 11 Angles  (Pages 205- 217)
• Chapter 12 Relative Sizes of Measurement Units  (Pages 219- 244)
• Chapter 13 Algebra: Perimeter and Area  (Pages 245- 258)

Grade 4 Homework FL. – Answer Keys

• Chapter 2 Multiply by 1-Digit Numbers Review/Test
• Chapter 3 Multiply 2-Digit Numbers Review/Test
• Chapter 4 Divide by 1-Digit Numbers Review/Test
• Chapter 5 Factors, Multiples, and Patterns Review/Test
• Chapter 6 Fraction Equivalence and Comparison Review/Test
• Chapter 7 Add and Subtract Fractions Review/Test
• Chapter 8 Multiply Fractions by Whole Numbers Review/Test
• Chapter 9 Relate Fractions and Decimals Review/Test
• Chapter 10 Two-Dimensional Figures Review/Test
• Chapter 11 Angles Review/Test
• Chapter 12 Relative Sizes of Measurement Units Review/Test
• Chapter 13 Algebra: Perimeter and Area Review/Test
• Chapter 1: Place Value, Multiplication, and Expressions
• Chapter 2: Divide Whole Numbers
• Chapter 3: Add and Subtract Decimals
• Chapter 4: Multiply Decimals
• Chapter 5: Divide Decimals
• Chapter 6: Add and Subtract Fractions with Unlike Denominators
• Chapter 7: Multiply Fractions
• Chapter 8: Divide Fractions
• Chapter 9: Algebra: Patterns and Graphing
• Chapter 10: Convert Units of Measure
• Chapter 11: Geometry and Volume
• Chapter 1: Divide Multi-Digit Numbers
• Chapter 2: Fractions and Decimals
• Chapter 3: Understand Positive and Negative Numbers
• Chapter 4: Model Ratios
• Chapter 5: Model Percents
• Chapter 6: Convert Units of Length
• Chapter 7: Exponents
• Chapter 8: Solutions of Equations
• Chapter 9: Independent and Dependent Variables
• Chapter 10: Area of Parallelograms
• Chapter 11: Surface Area and Volume
• Chapter 12: Data Displays and Measures of Center
• Chapter 13: Variability and Data Distributions

Go Math Answer Key for Grade 7

• Chapter 1: Adding and Subtracting Integers
• Chapter 2: Multiplying and Dividing Integers
• Chapter 3: Rational Numbers
• Chapter 4: Rates and Proportionality
• Chapter 5: Percent Increase and Decrease
• Chapter 6: Algebraic Expressions
• Chapter 7: Writing and Solving One-Step Inequalities
• Chapter 8: Modeling Geometric Figures
• Chapter 9: Circumference, Area, and Volume
• Chapter 10: Random Samples and Populations
• Chapter 11: Analyzing and Comparing Data
• Chapter 12: Experimental Probability
• Chapter 13: Theoretical Probability and Simulations
• Chapter 1 Real Numbers
• Chapter 2 Exponents and Scientific Notation
• Chapter 3 Proportional Relationships
• Chapter 4 Nonproportional Relationships
• Chapter 5 Writing Linear Equations
• Chapter 6 Functions
• Chapter 7 Solving Linear Equations
• Chapter 8 Solving Systems of Linear Equations
• Chapter 9 Transformations and Congruence
• Chapter 10 Transformations and Similarity
• Chapter 11 Angle Relationships in Parallel Lines and Triangles
• Chapter 12 The Pythagorean Theorem
• Chapter 13 Volume
• Chapter 14 Scatter Plots
• Chapter 15 Two-Way Tables

Give your kid the right amount of knowledge he needs as a part of your preparation by taking the help of our HMH Go Math Answer Key for Grades K-8. Resolve all your queries and assess your preparation standard using the Common Core Go Math Solution Key.

Practicing from the Go Math Answer Key for Grades K to 8 will provide a grade by grade roadmap and prepares students for College Readiness. Gradewise HMH Go Math Answer Key provided will develop problem-solving skills among students thereby helping them to Think, Explore and Grow. The Diverse Opportunities provided helps Kids to master the content with engaging activities.

Characteristics of Go Math Answer Key for Grades K to 8

Go through the below-listed features of referring to the HMH Go Math Anwer Key for Grades K to 8. They are outlined as follows

• All the Go Math Answer Key for Grades K to 8 are easy to download and we don’t charge any penny from you.
• Step by Step Solutions provided in the HMH Go Math Practice Key is aligned as per the College and Career Expectations.
• Solving from the Math 101 Practice Key helps you inculcate Higher Order Thinking Skills and you can answer any Question from your Homework, Assessment, or Review Test.
• More Rigorous Content made available meets the Common Core State Standards Initiative.
• You can gain a deeper knowledge of mathematical concepts and find solutions to all the Questions from Go Math Textbooks for Grades K, 1, 2, 3, 4, 5, 6, 7, 8

FAQs on Common Core HMH Go Math Answer Key

1. When Can I use the Go Math Answer Key for Grades K-8?

You can use the HMH Go Math Answer Key for Grades K to 8 while practicing the Go Math Textbook Questions as a part of your Homework or Assessment and make the most out of them.

2. Is there any site that provides the Common Core Go Math Solutions Key for Grades K, 1, 2, 3, 4, 5, 6, 7, 8?

Yes, you can find Go Math Answer Key for Grades K, 1, 2, 3, 4, 5, 6, 7, 8 all in one place i.e. ccssmathanswers.com a trusted and reliable portal.

3. Can I download HMH Go Math Answer Key PDF for free?

Yes, you can download the HMH Math 101 Practice Key for free on our page via quick links available and we don’t charge any amount for it.

• Big Ideas Math Answers

Elementary School Big Ideas Math Answers

• Big Ideas Math Answers Grade K
• Big Ideas Math Answers Grade 1
• Big Ideas Math Answers Grade 2
• Big Ideas Math Answers Grade 3
• Big Ideas Math Answers Grade 4
• Big Ideas Math Answers Grade 5

Middle School Big Ideas Math Solutions

• Big Ideas Math Answers Grade 6
• Big Ideas Math Answers Grade 6 Advanced
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High School Big Ideas Math Answers

• Big Ideas Math Algebra 1 Answers
• Big Ideas Math Algebra 2 Answers
• Big Ideas Math Geometry Answers
• Envision Math Common Core Grade K Answer Key
• Envision Math Common Core Grade 1 Answer Key
• Envision Math Common Core Grade 2 Answer Key
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• Envision Math Common Core Grade 4 Answer Key
• Envision Math Common Core Grade 5 Answer Key
• Envision Math Common Core Grade 6 Answer Key
• Envision Math Common Core Grade 7 Answer Key
• Envision Math Common Core Grade 8 Answer Key

Engage NY Eureka Math Answer Key

Engage NY Math Answer Key Pre K – 12

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• Eureka Math Pre K Answer Key
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• Math Expressions Grade 5 Homework and Remembering Answer Key
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• Math Expressions Grade K Homework and Remembering Answer Key