Loading...

## Solving Problems Using Rates## Video Lesson on How to Add and Subtract Rational Expressions## Solving Problems Using RatesIn this video lesson, learn how to set up rate problems so you can easily solve them. You will become familiar with the rate formula and you will see how easy it is to use. ## What Is a Rate Problem?What is a rate problem? A rate problem is a problem involving a rate of some sort, such as speed, earnings, etc. A rate is anything that can be gained or lost over time. You can gain speed over time and you can gain money over time too if you work hard. Rate problems usually come at you in the form of word problems. I know that word problems are the worst because you have to set up everything yourself. But let me show you how you can easily solve a rate word problem in this video. Keep watching!
So we are looking at our test paper and we see this problem: 'Two brothers want to purchase the next-generation entertainment system that costs $400. How many hours does the brother who makes $12 / hour have to work if the brother who makes $10 / hour can only work for 18 hours?'
See how useful solving rate problems can be? If you have a job and are getting paid on an hourly basis, you can use what you learn in this video lesson to figure out how much you have to work to purchase something you want. ## The Rate FormulaBefore we can set up our problem, I need to show you what our formula for rate is. Do you remember what rate means? A rate is anything that can be gained or lost over time. There are two ways you can write this mathematically. You can write it using division like so - rate = gain or loss/time - or, you can write it using multiplication like this - gain or loss = rate * time. This also happens to be the most often used form of this formula. Both are the same formula, they're just written differently. Your rate can be things like your speed or earnings. For our problem, we will use the formula with multiplication because it is the most commonly used form and it's easier for setting up our word problems. ## Setting up the ProblemSetting up the problem is the most important step in our solving process. I know, this is the hardest part of word problems, but like I said earlier, don't worry. I'm going to show you an easy way to do it. What we are going to do is make a table showing all of our important information for the problem. For the table, we are going to follow the format of our rate formula, the one that uses multiplication. Since we are making money, our formula is gain = rate * time. We will call our gain 'earnings' to make it clear that the amount we earn is our gain. We will have a separate column for each item. We will have a column for gain, a column for our equal sign, a column for rate, a column for our multiplication sign, and a column for the time. We will add an extra column in the beginning to label our brothers. We will call the brother who earns $10 / hour Brother 1 and the brother who earns $12 / hour Brother 2. Our table now looks like this. Rate formula set up Now that we have our table set up, we can fill it in with our information. We will use variables for the boxes we don't know or need to find out. The boxes under earnings are for the total amount earned over a period of time for each brother. We don't know this information, so we will put a variable einto the box for Brother 1. For Brother 2, we will put 400 - e because brother 2 only needs to make the difference between the cost of the entertainment system and the earnings of Brother 1. For the boxes in the rate column, we will put $10 for Brother 1 and $12 for Brother 2. Under the time column, we will put 18 for Brother 1 since the problem states he can only work for 18 hours. For Brother 2, we will put the variable t because we don't know how much time he needs to work yet. Look at that, we've filled out our table! That wasn't so bad, was it? Now that we have our table filled out, we can go ahead and solve the problem. |

Rate ready for solution ## Solving the ProblemBut what exactly is the problem asking for? Let's go back and see. 'Two brothers want to purchase the next-generation entertainment system that costs $400. How many hours does the brother who makes $12 / hour have to work if the brother who makes $10 / hour can only work for 18 hours?' Reading it carefully, I see that what it is asking for is the number of hours the brother who earns $12 / hour needs to work. Hmmm. That would be our Brother 2. The number of hours would be our variable t. So I need to solve for that variable. I go back to my table and look at it again. I see my t in the row for Brother 2. I also see an e in that row. Hmmm. What do I do about that e? Looking at Brother 1, I see that eequals 10 times 18. Aha! I can put that information in place for the e for Brother 2. This step I just did is called substitution. Writing it our mathematically, I get this - 400 - (10 * 18) = 12 * t. Hey! This looks like something I can solve using algebra skills. So let's see. I do my order of operations and multiply the 10 and the 18 first to get 180. Then I subtract that from the 400 to get 220. Then to solve for t, all I need to do is divide the 220 by 12. So then, my t would equal 220 / 12 = 18.33. Brother 2 would need to work 18.33 hours so the two brothers can purchase the next-generation entertainment system. Here's a word of caution. If you are given a multiple-choice test for this type of rate problem and the options didn't have 18.33 hours, but you had an 18 and a 19, you should pick 19 hours since most likely Brother 2 has to work full hours and not just part of an hour. For Brother 2 to be able to purchase the entertainment system, he would have to finish the partial hour to make 19 hours. If he only worked 18 hours, it wouldn't be enough. ## Lesson SummaryTo review, a rate problem is any problem involving rates. Rates are gains or losses occurring over time, such as speed or earnings. The formula for rate is your gain or loss equals your rate times time. In speed applications, your gain or loss is your distance. In earnings, your gain or loss is the total amount of earnings over a period of time. And remember, gains are positive and losses are negative. To solve rate word problems, you first set up a table following your rate formula. You fill in the table with the numbers you know and variables for the numbers you don't yet know. Make the variables fit the situation, like we did in the earnings column for the cost of our entertainment system. Then you solve your problem using algebra and substitutions. |

What are the Different Types of Numbers?
How to Build and Reduce Fractions
What is a Decimal Place Value
Statistical Analysis with Categorical Data
What is a Variable in Algebra?
What Are the Five Main Exponent Properties?
What is a Linear Equation?
What is an Absolute Value?
What are Polynomials, Binomials, and Quadratics?
How to Add and Subtract Rational Expressions
Perimeter of Triangles and Rectangles
Properties of Shapes: Rectangles, Squares and Rhombuses
What Is a Number Line?
Graph Functions by Plotting Points
What is a Linear Equation
How to Write Well: What Makes Writing Good?
What Are Literary Motifs?
How to Evaluate Reasoning
Reading Comprehension Techniques for the GMAT
How to Organize an Essay
How to Structure Sentences in an Essay
Using Rhetorical Skills to Write Better Essays
How to Write a Great Essay Quickly
GMAT Registration Information

- Five Main Exponent Properties
- Overview of the GRE Analytical Writing Measure
- Reading Comprehension Techniques for the GMAT
- Clocks
- Change of speech
- What is an Absolute Value?
- Smart GAT Study Plan
- Ordering of Sentence
- SAT Reading Section Structure, Patterns and Scoring
- Boats and Streams
- How to Solve a Rational Equation
- Direct and Indirect Speech
- Fractions
- Parallel, Perpendicular and Transverse Lines
- Coding - Decoding Tips and Tricks

All in this Category