# Solving Word Problems

To solve word problems, the information must be translated to the algebraic language.

### Common Algebraic Expressions

The double of a number: 2x

The triple a number: 3x

The quadruple of a number: 4x

Half of a number: x/2.

A third of a number: x/3.

A quarter of a number: x/4.

A number is proportional to 2, 3, 4, ...: 2x, 3x, 4x,...

A number to the square: x2

A number to the cube: x3

Two consecutive numbers: x and x + 1.

Two consecutive even numbers: 2x and 2x + 2.

Two consecutive odd numbers: 2x + 1 and 2x + 3.

Break 24 in two parts: x and 24 − x.

The sum of two numbers is 24: x and 24 − x.

The difference of two numbers is 24: x and 24 + x.

The product of two numbers is 24: x and 24/x.

The quotient of two numbers is 24; x and 24 · x.

### Age Word Problems

A father is 35 and his son, 5. After how many years is the father's age three times greater than the age of his son?

Years x

35 + x = 3 · (5 + x )

35 + x = 15 + 3 · x

20 = 2 · x    x = 10

After 10 years.

Today, John is three-quaters the age of his father and the difference in age is 15 years. Four years ago, the age of the father was twice the age of John. Find the age of the father and son 4 years ago.

John Father of John
Four years ago x 2x
Today x + 4 2x + 4

Age of John: 32 + 4 = 36.

Age or father: 2 · 32 + 4 = 68.

### Work Word Problems

Working together, it takes two workers 14 hours to complete a task. How long does it take to do the same task separately if one worker is twice as fast as the other?

Fast Slow
Time x 2x
Hours 1/x 1/2x

Fast 21 Hours

Slow 42 Hours

### Number Word Problems

If the double of a number is subtracted by its half and the result is 54. What is the number?

Tens (first figure) x + 1

There is a two-digit number and the digits that form it are consecutively ordered. The greater digit is in the first figure that forms the number (to the left) and the smaller digit is the second figure. The number equals six times the sum of its figures. What is the number?

Units (second figure) x

If there is a two-digit number, for example 65, it can be broken down, as follows: 6 · 10 + 5.

The two-digit number is (x +1) · 10 + x.

As this number is six times greater than the sum of its figures: x + x + 1 = 2x + 1, then:

(x +1) · 10 + x = 6 (2x + 1)

10x + 10 + x = 12 x + 6

10 x + x - 12x = 6 - 10

−x = −4       x = 4

Units 4

Tens 4 + 1 = 5

Number 54

### Geometric Word Problems

Find the value of the three angles in a triangle knowing that Angle B is 40° greater than Angle C and A is 40° greater than B.

C x

B x + 40

A x + 40 + 40 = x + 80

The three angles measure 180º.

x + x + 40 + x + 80 = 180;    x + x + x = 180 − 40 − 80;

3x = 60;    x= 20

C = 20º    ;B = 20º + 40º = 60º     A = 60º + 40º = 100º

The base of a rectangle is twice its height. What are its dimensions if the perimeter is 30 cm?

Height x

Base 2x

2 · x + 2 · 2x = 30    2x + 4x = 30      6x = 30    x = 5

Height 5 cm

Base 10 cm

### Mixture Word Problems

A trader has two types of coffee, the first is \$40/kg and the second, \$60/kg.

How many kilograms of each type of coffee must be mixed together to get 60 kilograms of a mixture that would cost \$50/kg?

1st Type 2nd Type Total
Number of kg x 60 − x 60
Value 40 · x 60 · (60 − x) 60 · 50

40x + 60 · (60 − x) = 60 · 50

40x + 3,600 − 60x = 3,000;    − 60x + 40x = 3,000 − 3,600;   20x = 600

x = 30;   60 − 30 = 30

The mix is 30 kilograms of the 1st type and 30 of the 2nd type

There are two different types of silver and each type is divided into portions of one gram. The first type is of 0.750 purity and the other of 0.950 purity. What is the weight of a bar that is formed by the contents of both types to obtain 1,800 grams of silver at 0.900 purity?

1st Type 2nd Type Total
No. of g x 1,800 − x 1,800
Silver 0.750 · x 0.950 · (1,800−x) 0.900 · 1,800

0.750 · x + 0.950 · (1,800−x) = 0.9 · 1,800

0.750 x + 1 710 − 0.950x = 1,620

0.750x − 0.950x = 1,620 − 1,710

−0.2x = − 90       x = 450

1st type 450 grams

2nd type 1,350 grams