Basic Math & Pre-Algebra For Dummies, 2nd Edition (2014)
Part III. Parts of the Whole: Fractions, Decimals, and Percents
Chapter 13. Word Problems with Fractions, Decimals, and Percents
In This Chapter
Adding and subtracting fractions, decimals, and percents in word equations
Translating the word of as multiplication
Changing percents to decimals in word problems
Tackling business problems involving percent increase and decrease
In Chapter 6, I show you how to solve word problems (also known as story problems) by setting up word equations that use the Big Four operations (adding, subtracting, multiplying, and dividing). In this chapter, I show you how to extend these skills to solve word problems with fractions, decimals, and percents.
First, I show you how to solve relatively easy problems, in which all you need to do is add or subtract fractions, decimals, or percents. Next, I show you how to solve problems that require you to multiply fractions. Such problems are easy to spot because they almost always contain the word of. After that, you discover how to solve percent problems by setting up a word equation and changing the percent to a decimal. Finally, I show you how to handle problems of percent increase and decrease. These problems are often practical money problems in which you figure out information about raises and salaries, costs and discounts, or amounts before and after taxes.
Adding and Subtracting Parts of the Whole in Word Problems
Certain word problems involving fractions, decimals, and percents are really just problems in adding and subtracting. You may add fractions, decimals, or percents in a variety of real-world settings that rely on weights and measures — such as cooking and carpentry. (In Chapter 15, I discuss these applications in depth.)
To solve these problems, you can use the skills that you pick up in Chapters 10 (for adding and subtracting fractions), 11 (for adding and subtracting decimals), and 12 (for adding and subtracting percents).
Sharing a pizza: Fractions
You may have to add or subtract fractions in problems that involve splitting up part of a whole. For example, consider the following:
· Joan ate of a pizza, Tony ate , and Sylvia ate . What fraction of the pizza was left when they were finished?
In this problem, just jot down the information that's given as word equations:
These fractions are part of one total pizza. To solve the problem, you need to find out how much all three people ate, so form the following word equation:
· all three = Joan + Tony + Sylvia
Now you can substitute as follows:
Chapter 10 gives you several ways to add these fractions. Here's one way:
However, the question asks what fraction of the pizza was left after they finished, so you have to subtract that amount from the whole:
Thus, the three people left of a pizza.
Buying by the pound: Decimals
You frequently work with decimals when dealing with money, metric measurements (see Chapter 15), and food sold by the pound. The following problem requires you to add and subtract decimals, which I discuss in Chapter 11. Even though the decimals may look intimidating, this problem is fairly simple to set up:
· Antonia bought 4.53 pounds of beef and 3.1 pounds of lamb. Lance bought 5.24 pounds of chicken and 0.7 pounds of pork. Which of them bought more meat, and how much more?
To solve this problem, you first find out how much each person bought:
You can already see that Antonia bought more than Lance. To find how much more, subtract:
· 7.63 – 5.94 = 1.69
So Antonia bought 1.69 pounds more than Lance.
Splitting the vote: Percents
When percents represent answers in polls, votes in an election, or portions of a budget, the total often has to add up to 100%. In real life, you may see such info organized as a pie chart (which I discuss in Chapter 17). Solving problems about this kind of information often involves nothing more than adding and subtracting percents. Here's an example:
In a recent mayoral election, five candidates were on the ballot. Faber won 39% of the vote, Gustafson won 31%, Ivanovich won 18%, Dixon won 7%, Obermayer won 3%, and the remaining votes went to write-in candidates. What percentage of voters wrote in their selection?
The candidates were in a single election, so all the votes have to total 100%. The first step here is just to add up the five percentages. Then subtract that value from 100%:
Because 98% of voters voted for one of the five candidates, the remaining 2% wrote in their selections.
Problems about Multiplying Fractions
In word problems, the word of almost always means multiplication. So whenever you see the word of following a fraction, decimal, or percent, you can usually replace it with a times sign.
When you think about it, of means multiplication even when you're not talking about fractions. For example, when you point to an item in a store and say, “I'll take three of those,” in a sense you're saying, “I'll take that one multiplied by three.”
The following examples give you practice turning word problems that include the word of into multiplication problems that you can solve with fraction multiplication.
Renegade grocery shopping: Buying less than they tell you to
When you understand that the word of means multiplication, you have a powerful tool for solving word problems. For instance, you can figure out how much you'll spend if you don't buy food in the quantities listed on the signs. Here's an example:
If beef costs $4 a pound, how much does of a pound cost?
Here's what you get if you simply change the of to a multiplication sign:
So you know how much beef you're buying. However, you want to know the cost. Because the problem tells you that 1 pound = $4, you can replace 1 pound of beef with $4:
Now you have an expression you can evaluate. Use the rules of multiplying fractions from Chapter 10 and solve:
This fraction reduces to . However, the answer looks weird because dollars are usually expressed in decimals, not fractions. So convert this fraction to a decimal using the rules I show you in Chapter 11:
At this point, recognize that $2.5 is more commonly written as $2.50, and you have your answer.
Easy as pie: Working out what's left on your plate
Sometimes when you're sharing something such as a pie, not everyone gets to it at the same time. The eager pie-lovers snatch the first slice, not bothering to divide the pie into equal servings, and the people who were slower, more patient, or just not that hungry cut their own portions from what's left over. When someone takes a part of the leftovers, you can do a bit of multiplication to see how much of the whole pie that portion represents.
Consider the following example:
Jerry bought a pie and ate of it. Then his wife, Doreen, ate of what was left. How much of the total pie was left?
To solve this problem, begin by jotting down what the first sentence tells you:
Doreen ate part of what was left, so write a word equation that tells you how much of the pie was left after Jerry was finished. He started with a whole pie, so subtract his portion from 1:
Next, Doreen ate of this amount. Rewrite the word of as multiplication and solve as follows. This answer tells you how much of the whole pie Doreen ate:
To make the numbers a little smaller before you go on, notice that you can reduce the fraction:
Now you know how much Jerry and Doreen both ate, so you can add these amounts together:
Solve this problem as I show you in Chapter 10:
This fraction reduces to . Now you know that Jerry and Doreen ate of the pie, but the problem asks you how much is left. So finish up with some subtraction and write the answer:
The amount of pie left over was .
Multiplying Decimals and Percents in Word Problems
In the preceding section, “Problems about Multiplying Fractions,” I show you how the word of in a fraction word problem usually means multiplication. This idea is also true in word problems involving decimals and percents. The method for solving these two types of problems is similar, so I lump them together in this section.
You can easily solve word problems involving percents by changing the percents into decimals (see Chapter 12 for details). Here are a few common percents and their decimal equivalents:
To the end: Figuring out how much money is left
One common type of problem gives you a starting amount — and a bunch of other information — and then asks you to figure out how much you end up with. Here's an example:
Maria's grandparents gave her $125 for her birthday. She put 40% of the money in the bank, spent 35% of what was left on a pair of shoes, and then spent the rest on a dress. How much did the dress cost?
Start at the beginning, forming a word equation to find out how much money Maria put in the bank:
· money in bank = 40% of $125
To solve this word equation, change the percent to a decimal and the word of to a multiplication sign; then multiply:
· money in bank = 0.4 × $125 = $50
Pay special attention to whether you're calculating how much of something was used up or how much of something is left over. If you need to work with the portion that remains, you may have to subtract the amount used from the amount you started with.
Because Maria started with $125, she had $75 left to spend:
· money left to spend
· = money from grandparents – money in bank
· = $125 – $50
· = $75
The problem then says that she spent 35% of this amount on a pair of shoes. Again, change the percent to a decimal and the word of to a multiplication sign:
· shoes = 35% of $75 = 0.35 × $75 = $26.25
She spent the rest of the money on a dress, so
· dress = $75 – $26.25 = $48.75
Therefore, Maria spent $48.75 on the dress.
Finding out how much you started with
Some problems give you the amount that you end up with and ask you to find out how much you started with. In general, these problems are harder because you're not used to thinking backward. Here's an example, and it's kind of a tough one, so fasten your seat belt:
Maria received some birthday money from her aunt. She put her usual 40% in the bank and spent 75% of the rest on a purse. When she was done, she had $12 left to spend on dinner. How much did her aunt give her?
This problem is similar to the one in the preceding section, but you need to start at the end and work backward. Notice that the only dollar amount in the problem comes after the two percent amounts. The problem tells you that she ends up with $12 after two transactions — putting money in the bank and buying a purse — and asks you to find out how much she started with.
To solve this problem, set up two word equations to describe the two transactions:
Notice what these two word equations are saying. The first tells you that Maria took the money from her aunt, subtracted some money to put in the bank, and left the bank with a new amount of money, which I'm calling money after bank. The second word equation starts where the first leaves off. It tells you that Maria took the money left over from the bank, subtracted some money for a purse, and ended up with $12.
This second equation already has an amount of money filled in, so start here. To solve this problem, realize that Maria spent 75% of her money at that time on the purse — that is, 75% of the money she still had after the bank:
I'm going to make one small change to this equation so you can see what it's really saying:
Adding 100% of doesn't change the equation because it really just means you're multiplying by 1. In fact, you can slip these two words in anywhere without changing what you mean, though you may sound ridiculous saying “Last night, I drove 100% of my car home from work, walked 100% of my dog, then took 100% of my wife to see 100% of a movie.”
In this particular case, however, these words help you to make a connection because 100% – 75% = 25%; here's an even better way to write this equation:
Before moving on, make sure you understand the steps that have brought you here.
You know now that 25% of money after bank is $12, so the total amount of money after bank is 4 times this amount — that is, $48. Therefore, you can plug this number into the first equation:
Now you can use the same type of thinking to solve this equation (and it goes a lot more quickly this time!). First, Maria placed 40% of the money from her aunt in the bank:
Again, rewrite this equation to make what it's saying clearer:
Now, because 100% – 40% = 60%, rewrite it again:
Thus, 0.6 × money from aunt = $48. Divide both sides of this equation by 0.6:money from aunt = $48 ÷ 0.6 = $80So Maria’s aunt gave her $80 for her birthday.
Handling Percent Increases and Decreases in Word Problems
Word problems that involve increasing or decreasing by a percentage add a final spin to percent problems. Typical percent-increase problems involve calculating the amount of a salary plus a raise, the cost of merchandise plus tax, or an amount of money plus interest or dividend. Typical percent decrease problems involve the amount of a salary minus taxes or the cost of merchandise minus a discount.
To tell you the truth, you may have already solved problems of this kind earlier in “Multiplying Decimals and Percents in Word Problems.” But people often get thrown by the language of these problems — which, by the way, is the language of business — so I want to give you some practice in solving them.
Raking in the dough: Finding salary increases
A little street smarts should tell you that the words salary increase or raise mean more money, so get ready to do some addition. Here's an example:
· Alison's salary was $40,000 last year, and at the end of the year, she received a 5% raise. What will she earn this year?
To solve this problem, first realize that Alison got a raise. So whatever she makes this year, it will be more than she made last year. The key to setting up this type of problem is to think of percent increase as “100% of last year's salary plus 5% of last year's salary.” Here's the word equation:
· this year’s salary = 100% of last year’s salary + 5% of last year’s salary
Now you can just add the percentages (see the nearby sidebar for why this works):
· this year’s salary = 105% of last year’s salary
Change the percent to a decimal and the word of to a multiplication sign; then fill in the amount of last year's salary:
· this year’s salary = 1.05 × $40,000
Now you're ready to multiply:
· this year’s salary = $42,000
So Alison's new salary is $42,000.
Earning interest on top of interest
The word interest means more money. When you receive interest from the bank, you get more money. And when you pay interest on a loan, you pay more money. Sometimes people earn interest on the interest they earned earlier, which makes the dollar amounts grow even faster. Here's an example:
Bethany placed $9,500 in a one-year CD that paid 4% interest. The next year, she rolled this over into a bond that paid 6% per year. How much did Bethany earn on her investment in those two years?
This problem involves interest, so it's another problem in percent increase — only this time, you have to deal with two transactions. Take them one at a time.
The first transaction is a percent increase of 4% on $9,500. The following word equation makes sense:
Now, substitute $9,500 for the initial deposit and calculate:
At this point, you're ready for the second transaction. This is a percent increase of 6% on $9,880:
Then subtract the initial deposit from the final amount:
So Bethany earned $972.80 on her investment.
Getting a deal: Calculating discounts
When you hear the words discount or sale price, think of subtraction. Here's an example:
Greg has his eye on a television with a listed price of $2,100. The salesman offers him a 30% discount if he buys it today. What will the television cost with the discount?
In this problem, you need to realize that the discount lowers the price of the television, so you have to subtract:
Thus, the television costs $1,470 with the discount.