Question

6. The ages of Hari and Harry are in the ratio 5:7. Four years from now the ratio of
their ages will be 3:4. Find their present ages.




7. The denominator of a rational number is greater than its numerator by 8. If the
numerator is increased by 17 and the denominator is decreased by 1, the number
obtained is
$\frac{3}{2} \: find \: the \: rational \: number$


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Answer 1

1st ǫuestion :

The ages of Hari and Harry are in the ratio 5:7. Four years from now the ratio of their ages will be 3:4. Find their present ages.

Given :

• Ratio of Ages of Hari and Harry = 5 : 7

After 4 years

• Ratio of ages of Hari and Harry = 3 : 4

Let present age of Hari and Harry be 5x and 7x

After 4 years their ages will be :

• Age of Hari = 5x + 4
• Age of Harry = 7x + 4

After 4 years ratio of their ages will be 3 : 4

⟶ 5x + 4/7x + 4 = 3/4

⟶ 4(5x + 4) = 3(7x + 4)

⟶ 20x + 16 = 21x + 12

⟶ 16 - 12 = 21x - 20x

⟶ 4 = x

So,

Present age of Hari :

= 5x

= 5 × 4

= 20 years.

Present age of Harry :

= 7x

= 7 × 4

= 28 years.

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2nd ǫuestion :

The denominator of a rational number is greater than its numerator by 8. If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is $\frac{3}{2}.$ Find the fraction.

Given :

• Denominator of Fraction is greater than it's numerator by 8.

Let the numerator of fraction be x so, Denominator of fraction is x + 8.

$\underline{ \large{ \mathscr{\dag\:A \bf{ccourding} \: to \: \mathscr {Q} \bf{uestion} ....}}}$

If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is 3/2.

⟶ x + 17/x + 8 - 1 = 3/2

⟶ x + 17/x + 7 = 3/2

⟶ 2(x + 17) = 3(x + 7)

⟶ 2x + 34 = 3x + 21

⟶ 34 - 21 = 3x - 2x

⟶ 13 = x

⟶ x = 13

So,

Numerator of fraction is 13

Denominator of fraction x + 8 = 21

Hence,

• Fraction = 13/21

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Answer 2

$\mathfrak{\huge{\purple{\underline{Solution \: 1}}}}$

$\large{\underline{\rm{\pink{\bf{Given:-}}}}}$

Ages of Hari and Harry are in the ratio = 5:7

Ratio of ages of Hari and Harry after 4 years = 3:4

$\large{\underline{\rm{\pink{\bf{To \: Find:-}}}}}$

The present age of Hari.

The present age of Harry.

$\large{\underline{\rm{\pink{\bf{Answer:-}}}}}$

Given that,

Ages of Hari and Harry are in the ratio = 5:7

Ratio of ages of Hari and Harry after 4 years = 3:4

According the question,

$\sf \dfrac{Age \: of \: Hari}{Age \: of \: Harry} =\dfrac{5}{7}$

Let Hari's age be 5x years and Harry's be 7x years.

Then after four years,

Hari's age = $\sf (5x+4)$ years

Harry's age = $\sf (7x+4)$ years

It's given that,

Ratio of ages of Hari and Harry after 4 years = 3:4

Then,

$\sf \dfrac{Hari's \: age \: after \: 4 \: years}{Harry's \: age \: after \: 4 \: years}=\dfrac{3}{4}$

$\sf \longrightarrow \dfrac{5x+4}{7x+4} =\dfrac{3}{4}$

$\sf 4(5x+4)=3(7x+4)$

$\sf 20x+16=21x+12$

$\sf 16-12=21x-20x$

$\sf x=4$

Therefore,

Present age of Hari = $\sf 5x=5 \times 4=20$ years

Present age of Harry = $\sf 7x=7 \times 4=28$ years

Hence the present age of Hari is 20 years and the present age of Harry is 28 years.

$\mathfrak{\huge{\purple{\underline{Solution \: 2}}}}$

$\large{\underline{\rm{\pink{\bf{To \: Find:-}}}}}$

The rational number.

$\large{\underline{\rm{\pink{\bf{Answer:-}}}}}$

Let the numerator of the rational number be x.

According to the question,

The denominator will x + 8

The rational number will be = $\sf \dfrac{x}{x} +8$

In this condition,

$\sf x+\dfrac{17}{x}+8-1=\dfrac{3}{2}$

$\sf = x+\dfrac{17}{x}+7=\dfrac{3}{2}$

$\sf = 2(x+17)=3(x+7)$

$\sf = 2x+34=3x+21$

$\sf =34-21=3x-2x$

$\implies \sf x=13$

Numerator of the rational number = 13

Denominator of the rational number = $\sf x + 8 = 13 + 8 = 21$

∴ Rational number = $\sf \dfrac{13}{21}$

$\mathfrak{\huge{\purple{\underline{Points \: to \: Note:-}}}}$

The expression on the left of the equality sign is the Left Hand Side (LHS). The expression on the right of the equality sign is the Right Hand Side (RHS)

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a variable.

Solving an equation means determining its root ie. to determine the value of the variable which satisfies it.