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

Common Algebraic Expressions

The double of a number: 2x

The triple of a number: 3x

The quadruple of a number: 4x

Half of a number: \frac { x }{ 2 }

A third of a number: \frac { x }{ 3 }

A quarter of a number: \frac { x }{ 4 }

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

A number to the square: { x }^{ 2 }

A number to the cube: { x }^{ 3 }

 

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 the two numbers will be 24 + x.

The difference between the two numbers will be 24 - x.

The product of the two numbers will be 24\times x.

The quotient of the two numbers will be \frac { 24 }{ x }.

 

Age Word Problems

Q. 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?

Let the number of years = x

35 + x = 3 \times (5 + x )

35 + x = 15 + 3x

20 = 2x

x = 10

After 10 years.

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Q. Today, John is three-quarters 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

\frac { 3 }{ 4 }(2x+4)=x+4+15

\frac { 6x+12 }{ 4 }=x+19

6x+12=4(x+19)

6x+12=4x+76

6x-4x=76-12

2x=64

x=32

Age of John: 32 + 4 = 36.

Age or father: (2 \times 32) + 4 = 68.

 

Work Word Problems

Q. 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 \frac { 1 }{ x } \frac { 1 }{ 2x }

 

\frac { 1 }{ x }+\frac { 1 }{ 2x }=\frac { 1 }{ 14 }

\frac { 2+1 }{ 2x }=\frac { 1 }{ 14 }

\frac { 3 }{ 2x }=\frac { 1 }{ 14 }

3 \times 14 = 2x

42 = 2x

x=21

For the slow worker=21 \times 2=42

 

Number Word Problems

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

2x-\frac { x }{ 2 }=54

\frac { 4x-x }{ 2 }=54

\frac { 3x }{ 2 }=54

3x=108

x=\frac { 108 }{ 3 }

x=36

 

Q. 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

Tens(first figure)x+1

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

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

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

(x +1) \times 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

Q. 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

 

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

Height x

Base 2x

(2 \times x) + (2 \times 2x) = 30

2x + 4x = 30

6x = 30

x = 5

Height 5㎝

Base 10㎝

Mixture Word Problems

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

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

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

40x + 60 \times (60 − x) = 60 \times 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

 

Q. There are two different types of silver and each type is divided into portions of one gram. The first type is 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 \times x 0.950 \times (1,800−x) 0.900 \times 1,800

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

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.