**Number Word Problems**

- There are two numbers whose sum is 72. One number is twice the other. What are the numbers?

Solution

Let x = smaller number

2x = larger number

Then

x + 2x = 72

3x = 72

x = 24

2x = 48

So

24 = smaller number

and

48 = larger number

- There are two numbers whose sum is 50. Three times the first is 5 more than twice the second. What are the numbers?

Solution

Let x = first number

50 – x = second number

**Then**

3x + 2(50 – x) + 5

3x = 100 – 2x + 5

5x = 105

x = 21 and 50 – x = 29

**So,**

The first number is 21 and the second number is 29.

- Separate 71 into two parts such that one part exceeds the other by 7. What are the numbers?

Solution

Let x = smaller part

71 – x = larger part

**Then**

(71 – x) – x = 7

71 – 2x = 7

-2x = 7 – 71

-2x = -64

x = 32 and 71 – x = 39

**So,**

The smaller part is 32 and the larger part is 39.

- Find three consecutive integers whose sum is 87.

**Solution**

Let x = first consecutive integer (smallest number)

x + 1 = second consecutive integer

x + 2 = third consecutive integer

**Equation**

x + (x + 1) + (x + 2) = 87

3x + 3 = 87

3x = 84

x = 28

x + 1 = 29

x + 2 = 30

**So,**

The three consecutive integers are 28, 29, and 30

**Check:**

The numbers 28, 29, and 30 are consecutive integers. The sum of 28, 29, and 30 is 87.

- Find three consecutive even integers such that the largest is three times the smallest.

Solution

Let x equal the first consecutive even integer.

Let x + 2 equal the second consecutive even integer.

Let x + 4 equal the third consecutive even integer.

**Equation**

The largest is 3 times the smallest.

x + 4 = 3x.

-2x = – 4

x = 2

x + 2 = 4

x + 4 = 6

**So,**

2, 4 and 6 are consecutive even integers and 6 is 3 times larger than 2.

- Four consecutive odd integers have a sum of 64. Find the integers.

Solution

Let x = first consecutive odd integer

x + 2 = second consecutive odd integer

x + 4 = third consecutive odd integer

x + 6 = fourth consecutive odd integer

**Equation**

The first integer + second integer + third integer + fourth integer = sum.

x + (x + 2) + (x + 4) + (x + 6) = 64

4x +12 = 64

4x = 52

x = 13

x + 2 = 15

x + 4 = 17

x + 6 = 19

**Check:**

The sum of 13, 15, 17, and 19 is 64.

- There is a number such that three times the number minus 6 is equal to 45. Find the number.

Solution

Let x = number

Three times the number minus 6 is 45

3x – 6 = 45

3x = 51

x = 17

- The sum of two numbers is 41.The larger number is 1 less than twice the smaller number. Find the numbers.

Solution

Let x = smaller number

41 – x = larger number

The larger number is twice the smaller number less 1.

**Then**

41 – x = 2x – 1

42 = 3x

x = 14

41 – x = 27

**So**

14 is the smaller number and 27 is the larger number.

- Separate 90 into two parts so that one part is four times the other part.

Solution

Let x = smaller part

90 – x = larger part (total minus x)

The larger part is four times the smaller part.

90 – x = 4x

-5x = -90

**Answers**

x = 18

90 – x = 72

**So,**

The two parts of 90 are 18 and 72.

- The sum of three consecutive integers is 54. Find the integers.

Solution

Let x = first consecutive integer

x + 1 = second consecutive integer

x + 2 = third consecutive integer

The sum of the three consecutive integers is 54.

x + (x + 1) + (x + 2) = 54

3x + 3 = 54

3x = 51

**Answers**

x = 17

x + 1 = 18

x + 2 = 19

**So,**

The 3 numbers are 17, 18 and 19

- There are two numbers whose sum is 53.Three times the smaller number is equal to 19 more than the larger number. What are the numbers?

Solution

Let x = smaller number

53 – x = larger number (total minus x)

Three times the smaller is the larger plus 19.

3x = (53 – x) + 19

4x = 72

**Answers:**

x = 18

53 – x = 35

**So,**

The two numbers are 18 and 35

- There are three consecutive odd integers. Three times the largest is seven times the smallest. What are the integers?

Solution

Let x = the first odd integer.

Let x + 2 = the second consecutive odd integer.

Let x + 4 = the third consecutive odd integer.

The difference between consecutive odd integers is 2. Three times the largest is seven times the smallest.

**Equation:**

3(x + 4) = 7x

3x +12 = 7x

-4x = -12

x = 3

**Answer**

x = 3

x + 2 = 5

x + 4 = 7

Three times the largest is seven times the smallest.

3(7) = 7(3)

**So,**

There are three consecutive odd integers are 3, 5 and 7.

- The sum of four consecutive even integers is 44. What are the numbers?

Solution

Let x = the first even integer.

Let x + 2 = the second even integer.

Let x + 4 = the third even integer.

Let x + 6 = the fourth even integer.

The difference between consecutive even integers is 2. The sum is 44.

**Equation**

x + (x + 2) + (x + 4) + (x + 6) = 44

4x + 12 = 44

4x = 32

x = 8

**Answer**

x = 8

x + 2 = 10

x + 4 = 12

x + 6 = 14

**Check the sum is 44.**

8 + 10 + 12 + 14 = 44

So the four consecutive even integers are 8, 10, 12 and 14.

- There are three consecutive integers. The sum of the first two is 35 more than the third. Find the integers.

Let x = first consecutive integer

x + 1 = second consecutive integer

x + 2 = third consecutive integer

The difference between consecutive integers is 1. The sum of the first two is the third plus 35.

**Equation**

x + (x + 1) = (x + 2) + 35

2x + 1 = x + 37

**Answer**

x = 36

x + 1 = 37

x + 2 + 38

So,

The three integers are 36, 37 and 38.

- A 25-foot-long board is to be cut into two parts. The longer part is 1 foot more than twice the shorter part. How long is each part?

Let x = shorter part

25 – x = longer part

**Equation**

25 – x = 2x + 1

-3x = -24

**Answer**

x = 8

25 – x = 17

**Alternate Solution **

Let x = shorter part

2x + 1 = longer part

**Equation**

x + 2x + 1 = 25

3x = 24

**Answer**

x = 8

2x + 1 = 17

**Check**

The sum is 25 and 8 + 17 = 25.

- Clara Nett went shopping for some canned goods which were on sale. She bought three times as many cans of tomatoes as cans of peaches and two times as many cans of tuna as cans of peaches. If Clara purchased a total of 24 cans, how many of each did she buy?

Let x = number of cans of peaches (smallest number)

3x = number of cans of tomatoes

2x = number of cans of tuna

**Equation **

x + 3x + 2x = 24

6x = 24

**Answer**

x = 4

3x = 12

2x = 8

**Check**

The sum is 24 and 4 + 12 + 8 = 24

- The first side of a triangle is 2 inches shorter than the second side. The third side is 5 inches longer than the second. If the perimeter of the triangle is 33 inches, how long is each side?

Let x = length of first side in inches

x + 2 = length of second side in inches

x + 2 + 5 = length of third side in inches

**Equation**

x + (x + 2) + (x + 2 + 5) = 33

2x + 2 + x + 7 = 33

3x + 9 = 33

3x = 24

**Answer**

x = 8

x + 2 = 10

x + 2 + 5 = 15

**Check**

The sum is 33 and 8 + 10 + 15 = 33

- In the afternoon, Calvin and Hobbes rode their bicycles 4 miles more than three times the distance in miles they rode in the morning on a trip to the lake. If the entire trip was 112 miles, how far did they ride in the morning and how far in the afternoon?

Let x= number of miles in the morning (smaller distance)

3x + 4 = number of miles in the afternoon

**Equation**

x + 3x + 4 = 112

4x + 4 = 112

4x = 108

**Answer**

x = 27

3x + 4 = 85

**Check**

The sum is 112 and 27 + 85 = 112.

- Mr. and Mrs. Luther and their son Martin own three cars. Martin drives 10 miles per week farther with his car than her father does with his. Mr. Luther drives twice as many miles per week as Mrs. Luther. If their total mileage per week is 160 miles, how many miles per week does each drive?

Let x = number of miles Mrs. Luther drives (smallest number)

2x = number of miles Mr. Luther drives

2x + 10 = number of miles Martin drives

**Equation**

x + 2x + (2x + 10) = 160

5x + 10 = 160

5x = 150

**Answer**

x = 30

2x = 60

2x + 10 = 70

**So,**

Mr. Luther drives 60, Mrs. Luther drives 30 and their son Martin drives 70.

**Check**

The sum is 160 and 30 + 60 + 70 = 160.

- There were 104,830 people who attended a rock festival. If there were 8110 more boys than girls, and 24,810 fewer adults over 50 years of age than there were girls, how many of each group attended the festival?

Let x = number of girls (smallest group)

x + 8110 = number of boys

x – 24810 = number of adults

Equation

x + x + 8110 + x – 24810 = 104,830

3x – 16700 = 104,830

3x = 121,530

**Answer**

x = 40,510

x + 8110 = 48,620

x – 24,810 = 15,700

**Check**

The total number of people is 104,830 and 40,510 + 48,620 + 15,700 = 104,830

21. In a 3-digit number, the hundreds digit is 4 more than the units digit and the tens digit is twice the hundreds digit. If the sum of the digits is 12, find the 3 digits. Write the number.

Let x = units digit (smallest number)

x + 4 = hundreds digit

2(x + 4) = tens digit

Equation

x + (x + 4) + 2(x + 4) = 12

x + x + 4 + 2x + 8 = 12

4x = 0

**Answer**

x = 0

x + 4 = 4

2(x + 4) = 8

The number is 480.

**Check**

The sum of the digits is 12 and 0 + 4 + 8 = 12.

- Seven times a number is the same as 12 more than 3 times the number. Find the number.

Let n = the number

7n = 3n + 12

4n = 12

n = 3

**Check**

7(3) = 3(3) + 12

- Six more than 5 times a number is the same as 9 less than twice the number. Find the number.

Let n = the number

5n + 6 = 2n – 9

3n + 6 = -9

3n = -15

n = -5

- Three less than 11 times a number is the same as the number decreased by 13. Find the number.

Let n = the number

11n – 3 = n – 13

10n – 3 = -13

10n = -10

n = -1

- One more than 3 times a number is the same as 5 times the number, decreased by 15. Find the number.

Let n = the number

3n + 1 = 5n – 15

1 = 2n – 15

16 = 2n

n = 8

- Twelve less than a number is the same as 6, decreased by 8 times the number. Find the number.

Let n = the number

n – 12 = 6 – 8n

9n – 12 = 6

9n = 18

n = 2

- Ten increased by 6 times a number is the same as 4 less than 4 times the number. Find the number.

Let n = the number

10 + 6n = 4n – 4

10 + 2n = -4

2n = -14

n = -7

- Eight times a number plus 3 times the number is the same as 9 more than 12 times the number. Find the number.

8n + 3n = 12n + 9

11n = 12n + 9

0 = n + 9

n = -9

- The sum of two numbers is 35. Three times the larger number is the same as 4 times the smaller number. Find the larger number.

(HINT: Let x = larger number and 35 – x = smaller number.)

3x = 4(35 – x)

3x = 140 – 4x

7x = 140

x = 20

So 20 is the larger number and 15 is the smaller number.

- The sum of two numbers is 24. Seven times the smaller number is the same as 5 times the larger number. Find the smaller number.

Let x = smaller number and 24 – x = larger number.

7x = 5(24 – x)

7x = 120 – ex

12x = 120

x = 10

So the smaller number is 10 and the larger is 14

**Check**

7(10) = 5(14)

70 = 70

- An orange has 20 fewer calories than a banana. If 7 bananas have the same number of calories as 9 oranges, how many calories are in a banana?

Let b = banana and r = orange

7b = 9r

7(r + 20) = 9r

7r + 140 = 9r

140 = 2r

r = 70

So the orange has 70 calories and the banana has 90 calories

- Keith weighs 20 kg more than Beth, while Henry weighs 30 kg less than twice as much as Beth. If Keith and Henry weigh the same, how much does Beth weigh (in kg)?

Let k = Keith

Let h = Beth

Let b = Henry

First equation k = b + 20

Second equation h + 30 = 2b

Third equation k = h

Substitute k from the third equation for the h in the second equation.

So then the first 2 equations are

First equation k = b + 20

Second equation k + 30 = 2b

Now substitute b + 20 from the first equation for k in the second equation.

b + 20 + 30 = 2b

b = 50

So Beth weighs 50 kg, Keith is 70 kg and Henry is also 70 kg

- Cycle Paths, inc. makes bicycles, tricycles, and unicycles. Last week they made 88 more bicycles than unicycles, and 5 times as many tricycles as unicycles. If they made 40 more bicycles than tricycles, how many unicycles did they make?

Let b = bicycle

Let t = tricycle

Let u = unicycle

First equation b = u + 88

Second equation t = 5u

Third equation t = 40 + u

Now substitute 5u from the second equation for t in the third equation.

5u = 40 + u

4u = 40

u = 10

So they made 10 unicycles, 98 bicycles and 50 tricycles.

- The second of three numbers is 8 more than the first. and the third number is 3 less than 3 times the first. If the third number is 15 more than the second, find the three numbers.

Let a = first number

Let b = second number

Let c = third number

First equation b = a + 8

Second equation c + 3 = 3a c = 3a – 3 solve for c

Third equation c = b + 15

Substitute 3a – 3 from the second equation in place c in the third equation giving

3a – 3 = b + 15

Next substitute a + 8 from the first equation for b in the above equation giving

3a – 3 = (3a – 3) + 15

Next solve for a

3a – 3 = 3a – 3 + 15

3a – 3 = 3a + 12

FIX

from the third equation

- In the championship game, Julius scored 5 points less than Kareem, and Wilt scored 1 point more than twice as many as Kareem. If Wilt scored 20 points more than Julius, how many points were scored by each player?
- Model Cars, Inc. makes red cars. white cars, and blue cars. The profit on a blue car is $10 more than the profit on a white car, and the profit on a red car is $7 less than the profit on a white car. If the profit on two red cars is $2 less than the profit on one blue car, what is the profit on a blue car?
- Last year Greg’s mother weighed 3 times as much as Greg. Since then Greg has gained 9 kg and his mother has lost 4 kg. Now his mother weighs only twice as much as Greg. Find their weights now.
- A hamburger costs 30 cents more than a hot dog and 20 cents less than a cheeseburger. If 3 hamburgers cost 75 cents less than 2 hot dogs and 2 cheeseburgers, how much does a hamburger cost?
- Smedly has just drawn two rectangles. The length of the first rectangle is 4 cm more than the length of the second. The width of the first is 9 cm; the width of the second is 6 cm. If the area of the first rectangle is 96 cm
^{2 }greater than the area of the second, find the length of each. - The second of two numbers is 7 times the first. Their sum is 72. Find the numbers.
- The larger of two numbers is 5 less than twice the smaller. Their sum is 43. Find the numbers.
- The sum of two numbers is 75. The first is 9 more than 5 times the second. Find the first number.
- Jack’s bowling score is 20 less than 3 times Jill’s score. The sum of their scores is 220. Find the score of each.
- Jennifer cut a board 2 m long into two pieces. One piece is 24 cm shorter than the other. Find the length of each piece.
- With optional equipment, an automobile cost $9120. If the cost of the basic car was $120 more than 4 times the price of the optional equipment, what was the cost of the basic car?
- The sum of three numbers is 207. The second number is 8 times the first, while the third is 3 less than the first. Find the numbers.
- The sum of three numbers is 161. The second number is 6 times the first, and the third is 5 more than the second. Find the numbers.
- A group of backpackers hiked 38 km over three days. The first day, they hiked 1 km less than 3 times as far as the second day. The third day, they hiked 2 km less than the first day. How far did they hike the first day?
- One week, Huey worked 7 hours less than Dewey, and Louie worked twice as long as Huey. Together they worked 87 hours. Find the number of hours worked by each.
- The sum of two numbers is 90. Their difference is 18. Find the numbers.
- The second of two numbers is 4 more than the first. The sum of the numbers is 56. Find the numbers.
- The number of girls at Sky High School is 60 greater than the number of boys. If there are 1250 students all together, how many girls are there?
- The second of two numbers is 5 more than twice the first. The sum of the numbers is 44. Find the numbers.
- The sum of two numbers is 75. The second number is 3 less than twice the first. Find the numbers.
- The larger of two numbers is 8 more than four times the smaller. If the larger is increased by four times the smaller, the result is 40. Find the numbers.
- The number of calories in a piece of pie is 20 less than three times the number of calories in a scoop of ice cream. The pie and ice cream together have 500 calories. How many calories are in each?
- The sum of two numbers is 4 less than twice the larger. If the larger is decreased by three times the smaller, the result is -20. Find the numbers.
- Three times the larger of two numbers is equal to four times the smaller. The sum of the numbers is 21. Find the numbers.
- The difference between two numbers is 16. Five times the smaller is the same as 8 less than twice the larger. Find the numbers.
- The larger of two numbers is 1 more than twice the smaller. The sum of the numbers is 20 less than three times the larger. Find the numbers.
- Two records and three tapes cost $31. Three records and two tapes cost $29. Find the cost of each record and each tape.
- The sum of two numbers is the same as four times the smaller number. If twice the larger is decreased by the smaller, the result is 30. Find the numbers.
- A group of students go out for lunch. If two have hamburgers and five have hot dogs, the bill will be $8.00. If five have hamburgers and two have hot dogs, the bill will be $9.50. What is the price of a hamburger?
- The price of a sweater is $5 less than twice the price of a shirt. If four sweaters and three shirts cost $200, find the price of each shirt and each sweater.
- A shipment of TV sets, some weighing 30 kg each and the others weighing 50 kg each, has a total weight of 880 kg. If there are 20 TV sets all together, how many weigh 50 kg?
- The second of two numbers is 4 times the first. Their sum is 50. Find the numbers.
- The larger of two numbers is 12 more than the smaller. Their sum is 84. Find the numbers.
- The sum of two numbers is 45. The first is 9 less than the second. Find the numbers.
- The second of two numbers is 5 more than twice the first. Their sum is 80. Find the numbers.
- The larger of two numbers is 1 less than 3 times the smaller. Their sum is 63. Find the numbers.
- Find two numbers whose sum is 92, if the first is 4 more than 7 times the second.
- The sum of two numbers is 172. The first is 8 less than 5 times the second. Find the
*first*number. - Together, a necklace and a bracelet cost $192. Find the price of each if the necklace costs 3 times as much as the bracelet.
- Grandpa’s age is 6 years less than 6 times Junior’s age. The sum of their ages is 78. Find each of their ages.
- The first of two films lasted 3 minutes less than twice as long as the second. Together the two films lasted 132 minutes. How long was the
*first*film? - The second of two numbers is 6 times the first. Their sum is 77. Find the numbers.
- The second of two numbers is 3 less than twice the first. Their sum is 36. Find the numbers.
- The sum of two numbers is 84. The first is 9 more than 4 times the second. Find the first number.
- The larger of two numbers is 1 less than 8 times the smaller. Their sum is 179. Find the numbers.
- An 84-meter length of cable is cut so that one piece is 18 meters longer than the other. Find the length of each piece.
- A bottle filled with water weighs 9.6 kilograms. If the water by itself weighs 5 times as much as the bottle, what is the weight of the bottle?
- Andy’s weight is 5 kilograms less than twice his brother’s. Together they weigh 100 kilograms. What are their weights?
- The sum of three numbers is 61. The second number is 5 times the first, while the third is 2 less than the first. Find the numbers.
- The sum of three numbers is 84. The second number is twice the first, and the third is 4 more than the second. Find the numbers.
- Together a chair, a table, and a lamp cost $562. The chair costs 4 times as much as the lamp, and the table costs $23 less than the chair. Find the cost of the table.
- The sum of the angle measures of any triangle is 180°. Find the angle measures of a triangle if the second angle measures 100 less than twice the first, and the third angle measures 25° more than the second.