Basic Math and Pre-Algebra For Dummies (67 page)

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.

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.

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.