Practice solving equations with 1 variable

**1. A number is increased by 2 and then multiplied by 3. The result is 24. What is this number?**

A) 4

B) 6

C) 8

D) 10

**2. My father’s age divided by 5 is equal to my brother’s age divided by 3. My brother is 3 years older than me. My father’s age is 3 less than 2 times my age. How old is my father?**

A) 34

B) 45

C) 56

D) 61

**3. (x – 2) / 4 – (3x + 5) / 7 = -3, x = ?**

A) 6

B) 7

C) 10

D) 13

**4. 1 / (1 + 1 / (1 – 1/x)) = 4, x = ?**

A) -3/4

B) 3/7

C) 4/7

D) 3/4

**5. Jane is three times Peter’s age. When Peter is at Jane’s age, Jane will be 8 less than 2 times Peter’s age. Find the sum of Jane and Peter’s age at present.**

A) 18

B) 21

C) 25

D) 32

## Answer Key

**1. B**

Let us call this number by x:

This number is increased by 2: x + 2

Then, it is multiplied by 3: 3(x + 2)

The result is 24: 3(x + 2) = 24 … Solving this linear equation, we obtain the value of the number:

x + 2 = 24 / 3

x + 2 = 8

x = 8 – 2

x = 6

**2. B**

My age: x

My brother is 3 years older than me: x + 3

My father is 3 less than 2 times my age: 2x – 3

My father’s age divided by 5 is equal to my brother’s age divided by 3: (2x – 3) / 5 = (x + 3) / 3

By cross multiplication:

5(x + 3) = 3(2x – 3)

5x + 15 = 6x – 9

x = 24

My father’s age: 2.24 – 3= 48 – 3 = 45

**3. C
**There are two fractions containing x and the denominators are different. First, let us find a common denominator to simplify the expression. The least common multiplier of 4 and 7 is 28. Then,

7(x – 2) / 28 – 4(3x + 5) / 28 = – 3.28 / 28 … Since both sides are written on the denominator 28 now, we can eliminate them:

7(x – 2) – 4(3x + 5) = -84

7x – 14 – 12x – 20 = -84

-5x = -84 + 14 + 20

-5x = -50

x = 50/5

x = 10

**4. B
**We can follow outside to inside method to solve this type of problems. x is in the inner part of this fraction; then, we need to narrow the circle to reach x:

1 / (1 + 1 / (1 – 1/x)) = 4

This means that (1 + 1 / (1 – 1/x)) is equal to 1/4. Then,

1 + 1 / (1 – 1/x) = 1/4

1 / (1 – 1/x) = 1/4 – 1

1 / (1 – 1/x) = – 3/4

This means that 1 – 1/x = – 4/3. Then,

1 – 1/x = – 4/3

1 + 4/3 = 1/x

1/x = 7/3

So, x = 3/7.

**5. D**

Let us say that Peter is “a” years old today. Then, Jane is “3a”. When Peter is “3a”, 3a – a = 2a years will have passed and so, Jane will be 3a + 2a = 5a. Then, we are given that Jane will be 8 less than 2 times Peter’s age. So;

5a = 2^{.}3a – 8

5a = 6a – 8

a = 8.

We are asked to find the sum of Jane and Peter’s age at present.

Today, Peter is a = 8 years old and Jane is 3a = 24 years old.

Their sum is 8 + 24 = 32.

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