Question

The numerator of a fraction is one less than its denominator. If 3 is added to each of the numerator and denominator, the fraction is increased by 3/28, find the fraction. 28

Answer 1

Answer:

★ Fraction = 3/4 ★

Step-by-step explanation:

Given:

• Numerator of a fraction is one less than it's denominator.
• If 3 is added to numerator and denominator then fraction is increased by 3/28

To Find:

• What is the fraction?

Solution: Let the denominator be x therefore numerator will be (x – 1)

∴ Fraction: = Numerator/Denominator

→ Original Fraction = (x–1/x)

★ After adding 3 to numerator and denominator★

• Numerator= x–1+3= x+2
• Denominator= x+3

∴ New Fraction = Numerator/Denominator

→ Fraction = (x+2/x+3)

According to the question:

$\small\implies{\sf }$ x+2/x+3 = x–1/x + 3/28 [ Take LCM ]

$\small\implies{\sf }$ x+2/x+3 = 28x – 28 + 3x/28x

$\small\implies{\sf }$x+2/x+3 = 31x – 28/28x

•By Cross Multiplication •

$\small\implies{\sf }$ 28x (x+2) = 31x – 28 (x+3)

$\small\implies{\sf }$ 28x² + 56x = 31x (x+3) –28 (x+3)

$\small\implies{\sf }$ 28x² + 56x = 31x² + 93x –28x – 84

$\small\implies{\sf }$ 28x² + 56x = 31x² + 65x – 84

$\small\implies{\sf }$ 0 = 31x² – 28x² + 65x – 56x – 84

$\small\implies{\sf }$ 0 = 3x² + 9x – 84

→ 3x² + 9x – 84 [ Divide by 3 ]

$\small\implies{\sf }$ 3x²/3 + 9x/3 – 84/3

$\small\implies{\sf }$ x² + 3x – 28

$\small\implies{\sf }$ x² + 7x – 4x – 28 [ By middle term splitting ]

$\small\implies{\sf }$ x (x + 7) – 4 (x + 7)

$\small\implies{\sf }$ (x + 7) (x – 4)

Here x = 7 and –4 { x is a natural number x > 0 }

∴ x = 7 because x cannot be negative

Therefore, Fraction will be (x–1/x)

★ Putting the value of x ★

→ Fraction = Numerator/ Denominator

→ Fraction = (4 – 1/4)

→ Fraction = 3/4

Hence, The fraction is 3/4