**Solving multi step word problems :**

To solve some word problems, we may have to do more than one step. There we will be using more than one binary operation to solve the problems.

**Example 1 :**

The denominator of a fraction exceeds the numerator by 5. If 3 be added to both, the fraction becomes 3/4. Find the fraction.

**Solution : **

**Step 1 :**

Let "x" be the numerator.

"The denominator of the fraction exceeds the numerator"

From the above information,

fraction = x / (x+5) ------(1)

**Step 2 :**

"If 3 be added to both, the fraction becomes 3/4"

From the above information, we have

(x+3) / (x+5+3) = 3/4

**Step 3 :**

(x+3)/(x+8) = 3/4 ---------> 4(x+3) = 3(x+8)

4x + 12 = 3x + 24 --------> x = 12

**Step 4 :**

(1)--------> x / (x+5) = 12 / (12+5) = 12/27

Hence, the required fraction is 12/27

**Example 2 :**

If thrice of A's age 6 years ago be subtracted from twice his present age, the result would be equal to his present age. Find A's present age.

**Solution :**

**Step 1 :**

Let "x" be A's present age.

A's age 6 years ago = x - 6

**Step 2 :**

Thrice of A's age 6 years ago = 3(x-6) --------(1)

**Step 3 :**

Twice his present age = 2x --------(2)

**Step 4 : **

According to the question, (2) - (1) = A's present age

2x - 3(x-6) = x ----> 2x - 3x + 18 = x ----> 18 = 2x

18 = 2x --------> 9 = x

Hence, A's present age is 9 years

**Example 3 :**

A number consists of two digits. The digit in the tens place is twice the digit in the units place. If 18 be subtracted from the number, the digits are reversed. Find the number.

**Solution :**

**Step 1 : **

Let "x" be the digit in units place.

Then the digit in the tens place = 2x

**Step 2 :**

According to the question, (2x)x - 18 = x(2x)

(2x)x - 18 = x(2x) -----> (2x).10 + x.1 - 18 = x.10 + (2x).1

20x + x - 18 = 10x + 2x -------> 21x - 18 = 12x

21x - 18 = 12x ------> 9x = 18 ------> x = 2

**Step 3 :**

So, the digit at the units place = x = 2

and the digit at the tens place = 2x = 2(2) = 4

Hence the required number is 42

**Example 4 :**

The width of the rectangle is 2/3 of its length. If the perimeter of the rectangle is 80 cm. Find its area.

**Solution :**

**Step 1 : **

Let "x" be the length of the rectangle.

Then, width of the rectangle = (2/3)x

**Step 2 : **

Perimeter = 80 cm ----> 2(l + w) = 80 ----> l + w = 40

l + w = 40 ----> x + (2/3)x = 40 ----> (3x+2x) / 3 = 40

(3x+2x) / 3 = 40 ----> 5x = 120 ----> x = 24

So, length = x = 24 cm

**Step 3 : **

Width = (2/3)x = (2/3)24 = 16 cm

**Step 4 :**

Area = l x w = 24x16 = 384 square cm.

Hence, area of the rectangle is 384 square cm

**Example 5 :**

The ratio of the no. of boys to the no. of girls in a school of 720 students is 3:5. If 18 new girls are admitted in the school, find how many new boys may be admitted so that the ratio of the no. of boys to the no. of girls may change to 2:3.

**Solution :**

**Step 1 : **

Sum of the terms in the given ratio = 3 + 5 = 8

So, no. of boys in the school = 720 x (3/8) = 270

No. of girls in the school = 720 x (5/8) = 450

**Step 2 : **

Let "x" be the no. of new boys admitted in the school.

No. of new girls admitted = 18 (given)

After the above new admissions,

no. of boys in the school = 270+x

no. of girls in the school = 450+18 = 468

**Step 3 :**

The ratio after the new admission is 2 : 3 (given)

So, we have

(270 + x) : 468 = 2 : 3

3(270 + x) = 468 x 2

(using cross product rule in proportion)

810 + 3x = 936

3x = 126

x = 42

Hence the no. of new boys admitted in the school is 42

After having gone through the stuff given above, we hope that the students would have understood "Solving multi step word problems".

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