## Solving Word Problems with Arithmetic Expressions

Sometimes math tests show you HOW a problem could be solved instead of asking you to pick the actual answer. This can be especially tricky if you solved the problem a different way (and most problems do have several ways to arrive at the same correct answer).

For example, let say that you want to find out how much Pat and Chris earned last week. Pat worked 30 hours and is paid \$7 per hour. Chris is paid \$8 per hour and worked 25 hours.

Did you notice that the info was flip-flopped? (That can make it more difficult to match things up, but --take a deep breath-- it can still be done!!)

• First, think about what you're trying to find... total pay for BOTH people.
• So you'll need one person's pay PLUS the other person's pay.
• Pat worked 30 hours and is paid \$7 per hour -- you could add \$7 over and over 30 times (or just multiply 30 times 7).
• Chris is paid \$8 per hour and worked 25 hours -- you could add \$8 over and over 25 times (or just multiply 25 times 8). By the way, that's the same as saying, "Chris worked 25 hours and is paid \$8 per hour" (in case it's easier for you to think of this with identical expressions).
• So Pat earned (30 * 7) dollars AND Chris earned (25 * 8) dollars
• (30 *7) + (25 * 8)

Don't panic!! You can either calculate the answer yourself (and then see which answer choice yields the same amount) or think about the steps you'd go through to solve the problem and see which of the choices matches that procedure.

Read the word problem carefully; then decide which arithmetic expression would yield the answer to the question.

#### Sample Problem #1:

How much would you pay if you bought:   2 items that cost \$5 each and 3 items that cost \$4 each

Which expression shows the total cost?

A)   (2 * 3) + (5 * 4)
B)   (2 + 5) * (3 + 4)
C)   (2 * 5) + (3 * 4)
D)   (2 + 3) * (5 + 4)

##### Find the letter of the answer you chose above:

A - That's items times items PLUS cost times cost.

B - That's item plus cost TIMES item plus cost.

C - That's items times cost PLUS items times cost.

D - That's items plus items TIMES cost plus cost.

You WANTED the quantity of the first type of item times the cost of each one PLUS the quantity of the second type of item times the cost of each of them:

 2 items that cost \$5 each and 3 items that cost \$4 each 2 times \$5 + 3 times \$4 2 * 5 + 3 * 4 10 + 12 \$22 (choice "C").

Sometimes you may find it helpful to actually calculate the answer and then work backwards from the expressions to find the one that yields the same amount.

#### Sample Problem #2:

How far would you travel if you drove:   2 hours at 50 miles per hour and 3 hours at 40 miles per hour

Which expression shows the total distance traveled?
A)   (2 * 40) + (3 * 50)
B)   (2 * 50) + (3 * 40)
C)   (2 + 3) * (50 + 40)
D)   (2 + 50) * (3 + 40)

#### Distance equals Rate times Time D = R * T

##### Find the letter of the answer you chose above:

A - That's 2 hours times 40 mph PLUS 3 hours times 50 mph.

B - That's 2 hours times 50 mph PLUS 3 hours times 40 mph.

C - That's time (in hours) plus time (in hours) TIMES rate plus rate.

D - That's time (in hours) plus rate TIMES time (in hours) plus rate.

You WANTED the number of hours traveled at the first rate of speed times the distance per hour PLUS the number of hours traveled at the second rate of speed times the distance per hour:
 2 hours at 50 miles per hour and 3 hours at 40 miles per hour 2 hours at 50 mph and 3 hours at 40 mph 2 times 50 + 3 times 40 2 * 50 + 3 * 40 100 miles + 120 miles 220 miles (choice "B").
You may sometimes find it helpful to actually calculate the answer for each expression (while keeping in mind what each amount represents); then see which approach "makes sense" in solving the problem.

1. Betty drove for six hours at 45 miles per hour and for two hours at 50 miles per hour. Which expression shows the total distance she drove? AND How many miles did she drive?
1. 6(45 + 50)
2. (45 + 50) / 8
3. (6 * 45) + (2 * 50)
4. (6 * 45) - (2 * 50)

2. Perian bought four pounds of cheese at \$3 a pound and six pounds of ham at \$2 a pound. Which expression shows the total cost? AND How much did Perian's purchases cost in all?
1. (3 * 2) + (4 * 6)
2. (4 * 3) + (6 * 2)
3. (4 + 3) * (6 + 2)
4. (6 * 2) * (4 * 3)

3. Kim worked six days last week and earned \$60 each day; this week she worked four days and earned \$75 each day. Which expression shows how much more she earned last week than this week? How much more did Kim earn last week?
1. (6 * 60) - (4 * 75)
2. (6 - 4) * (75 - 60)
3. (60 - 6) * (75 - 4)
4. (6 * 60) + (4 * 75)

 Answers: 1-c (370 miles) 2-b (\$24 in all) 3-a (\$60 more)

For more practice, see: Solving Word Problems