Solving Word Problems with Simple Equations

For these word problems:

a) Make a "let" statement to name the unknown or variable,
b) Translate the words into an equation in that variable,
c) Solve the problem and give your answer in words (with units).

1) A number increased by 35 is 51. What is that number? ( solution )

2) One number is eight times a smaller one. If the smaller number is subtracted
from the larger one, the remainder is 21. Find the two numbers. ( solution )

3) When 32 is subtracted from 7 times a number, we get 24. What is the number? ( solution )

4) If 12 times a number be increased by 5, the sum is 25 more
than 7 times the number. What is the number? ( solution )

5) The length of a lot is 5 times its width. The perimeter is 300ft.
What are the length and width of this lot? ( solution )

6) Frank is 5 years older than Fred. The sum of their ages is 25.
How old is each of them? ( solution )

7) Five less than 10 times a number, is the same as
the sum of 19 and 8 times the number. What is the number? ( solution )

8) Our hockey team played 152 games this season.
They won 30 games more than they lost. How many games did they win? ( solution )

9) The sum of two consecutive integers is 113. What are the integers? ( solution )

10) Find three consecutive integers whose sum is 114. ( solution )

SOLUTIONS

1) a number increased by 35 is 51. What is that number?

a) Let n equal the number.
b) the equation is: n + 35 = 51
c) we subtract 35 from both sides of the = to get n = 51 – 35 = 16. The number is 16.

2) One number is eight times a smaller one. If the smaller number is subtracted
from the larger one, the remainder is 21. Find the two numbers.

a) Let n equal the smaller number. Then 8n = the other one.
b) the equation is: 8nn = 21.
c) combine like terms; 7n = 21 -- divide by 7; n = 3. The numbers are 3 and 24.

3) When 32 is subtracted from 7 times a number, we get 24. What is the number?

a) Let n equal the number.
b) the equation is: 7n – 32 = 24
c) we add 32 to both sides, then divide by 7: 7n = 56 making n = 8. The number is 8.

4) If 12 times a number be increased by 5, the sum is 25 more
than 7 times the number. What is the number?

a) Let n equal the number.
b) the equation is: 12n + 5 = 25 + 7n.
c) transpose + 5 and 7n; 5n = 20; divide by 5; so n = 4. The number is 4.

5) The length of a lot is 5 times its width. The perimeter is 300ft.
What are the length and width of this lot?

a) Let W = the Width of the lot, so 5W = the Length.
b) the equation is: 2(5W + W) = 300.
c) divide through by 2; collect terms: 6W = 150; divide by 6. W = 25.
The Width measures 25 ft. so the length measures 5 × 25 or 125 ft.

6) Frank is 5 years older than Fred. The sum of their ages is 25.
How old is each of them?

a) Let F = Fred's age. That makes Frank's age F + 5.
b) the equation is: F + (F + 5) = 25.
c) remove brackets, collect terms to get 2F + 5 = 25; transpose, divide by 2; F = 10.
Fred and Frank are 10 and 15years old respectively.

7) Five less than 10 times a number is the same as
the sum of 19 and 8 times the number. What is the number?

a) Let n equal the number.
b) the equation is: 10n – 5 = 19 + 8n
c) we transpose –5 and 8n to get 2n = 24, which makes n = 12. The number is 12.

8) Our hockey team played 152 games this season.
They won 30 games more than they lost. How many games did they win?

a) Let x be the number of games they lost. That means they won x + 30 games.
b) the equation is games won + games lost = total games: x + x + 30 = 152
c) collect the x's, transpose 30, to get 2x = 122 making x = 61. This makes x + 30 = 91.
The team won 91 games.

9) The sum of two consecutive integers is 113. What are the integers?

a) Let x be the first integer (whole number) so x + 1 is the next integer.
b) the equation is: x + x + 1 = 113.
c) we get 2x = 112 making x = 56 and x + 1 = 57. The integers are 56 and 57.

10) Find three consecutive integers whose sum is 114.

a) Let x be the first integer, so (x + 1) and (x + 2) are the next 2 integers.
b) the equation is: x + (x + 1) + (x + 2) = 114.
c) simplifying we get: 3x + 3 = 114 which means x = 37. The integers are 37, 38, 39.

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