**Substitution Word Problems :**

In this section, we will learn how to solve word problems using substitution.

**Example 1 : **

One number is greater than thrice the other number by 2. If four times the smaller number exceeds the greater by 5, find the numbers.

**Solution : **

Let x be the smaller number and y be the greater number.

**Given : **One number is greater than thrice the other number by 2.

Then, we have

y = 3x + 2 -----(1)

**Given : **Four times the smaller number exceeds the greater by 5.

Then, we have

4x = y + 5

Substitute (3x + 2) for y.

4x = 3x + 2 + 5

4x = 3x + 7

Subtract 3x from each side.

x = 7

Substitute 7 for x in (1).

(1)-----> y = 3(7) + 2

y = 21 + 2

y = 23

So, the numbers are 7 and 23.

**Example 2 : **

The ratio of income of two persons is 1 : 2 and the ratio of their expenditure is 3 : 7. If the first person saves $2000 and the second person saves $3000, find the expenditure of each person.

**Solution : **

From the ratio of income 1 : 2, the incomes of the two persons can be assumed as x and 2x.

From the ratio of expenditure 3 : 7, the expenditures of the two persons can be assumed as 3y and 7y.

**Savings = Income - Expenditure**

The first person saves $3000.

Then, we have

x - 3y = 3000

Add 2y to each side.

x = 3000 + 3y -----(1)

The second person saves $7000.

Then, we have

2x - 7y = 7000

Substitute (3000 + 3y) for x.

(1)-----> 2(2000 + 3y) - 7y = 3000

4000 + 6y - 7y = 3000

4000 - y = 3000

Subtract 1000 from each side.

-y = -1000

Multiply each side by -1.

y = 1000

3y = 3(1000) = 3000

7y = 7(1000) = 7000

So, the expenditures of the two persons are $3000 and $7000.

**Example 3 : **

Three chairs and two tables cost $700 and five chairs and six tables cost $1100. Find the cost of each chair.

**Solution : **

Let x and y be the costs of each chair and table respectively.

Then, we have

3x + 2y = 700 -----(1)

5x + 6y = 1700 -----(2)

Multiply (1) by 3.

(1) ⋅ 3 -----> 9x + 6y = 2100

Subtract 9x from each side.

6y = 2100 - 9x

Substitute (2100 - 9x) for 6y in (2).

(2)-----> 5x + 2100 - 9x = 1700

2100 - 4x = 1700

Subtract 2100 from each side.

-4x = -400

Divide each side ny -4.

x = 100

So, the cost of each chair is $100.

**Example 4 : **

A father's age is three times the sum of the age of his two son. After 5 years, his age will be twice the sum of the ages of his two sons. Find the present age of the father.

**Solution : **

Let x be the present age of the father and y be the sum of the present ages of the two sons.

At present :

x = 3y -----(1)

After 5 years :

x + 5 = 2(y + 5 + 5)

x + 5 = 2(y + 10)

x + 5 = 2y + 20

Subtract 5 from each side.

x = 2y + 15

Substitute 3y for x.

3y = 2y + 15

Subtract 2y from each side.

y = 15

Substitute 15 for y in (1).

(1)-----> x = 3(15)

x = 45

So, the present age of the father is 45 years.

**Example 5 : **

If the numerator of a fraction is increased by 2 and the denominator by 1, it becomes 1. Again if the numerator is decreased by 4 and the denominator by 2, it becomes 1/2. Find the fraction.

**Solution : **

Let x/y be the required fraction.

**Given : **If the numerator of a fraction is increased by 2 and the denominator by 1, it becomes 1.

(x + 2) / (y + 1) = 1

Multiply each side by (y + 1).

x + 2 = 1(y + 1)

x + 2 = y + 1

Subtract 2 from each side.

x = y - 1 -----(1)

**Given : **If the numerator is decreased by 4 and the denominator by 2, it becomes 1/2.

(x - 4) / (y - 2) = 1 / 2

Multiply each side by 2(y - 2).

2(x - 4) = 1(y - 2)

2x - 8 = y - 2

Add 8 to each side.

2x = y + 6

Substitute (y - 1) for x.

2(y - 1) = y + 6

2y - 2 = y + 6

Add 2 to each side.

2y = y + 8

Subtract y from each side.

y = 8

Substitute 8 for y in (1).

(1)-----> x = 8 - 1

x = 7

Therefore, we have

x / y = 7 / 8

So, the required fraction is 7/8.

After having gone through the stuff given above, we hope that the students would have understood, "Substitution Word Problems".

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Widget is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Time and work word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**