Question

Q13. The ages of Heena and Meena are in the ratio 5:3.
After 6 years, their ages will be in
the ratio 7:5. Find their present ages.​

Answer 1

Answer :-

$\\\;\underbrace{\underline{\sf{Question's\;\;Analysis\;:-}}}$

Here the concept of Linear Equations has been used. We see that in both cases the ratio of ages has been given. So surely their must a constant x by which both the ratios shall be multiplied to get the original values. Using this concept,

Let's do it.

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★ Equation Used :-

$\\\;\boxed{\sf{\dfrac{5x\;+\;6}{3x\;+\;6}\;=\;\bf{\dfrac{7}{5}}}}$

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★ Solution :-

Given,

» Present ratio of ages of Heena and Meena = 5 : 3

» After six years, final ratio of ages of Heena and Meena = 7 : 5

• Let the constant be x by which both the numerator and denominator of initial ratio should be multiplied. Then,

$\\\;\sf{\Longrightarrow\;\;\;Present\;ratio\;of\;ages\;of\;Heena\;and\;Meena\;=\;\bf{\dfrac{5x}{3x}}}$

$\\\;\sf{\Longrightarrow\;\;\;Present\;age\;of\;Heena\;\;=\;\bf{5x}}$

$\\\;\sf{\Longrightarrow\;\;\;Present\;age\;of\;Meena\;\;=\;\bf{3x}}$

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~ For the value of x :-

Since the ratio of ages after 6 years is 7 : 5 .

Then their will be addition of 6 years in ratio of present ages.

So,

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;\dfrac{(5x\;+\;6)}{(3x\;+\;6)}\;=\;\bf{\dfrac{7}{5}}}$

By cross - multiplication, we get,

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;5(5x\;+\;6)\;=\;\bf{7(3x\;+\;6)}}$

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;25x\;+\;30\;=\;\bf{21x\;+\;42}}$

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;25x\;+\;21x\;=\;\bf{42\;-\;30}}$

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;4x\;=\;\bf{12}}$

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;x\;=\;\bf{\dfrac{12}{4}}}$

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;x\;=\;\bf{3}}$

$\\\;\underline{\boxed{\tt{Hence,\;\;Value\;\;of\;\;x\;=\;\bf{3}}}}$

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~ For the present ages of Heena and Meena :-

Clearly we can find the value of both of their ages, by applying values of x.

$\\\;\sf{:\mapsto\;\;\;Present\;age\;of\;Heena\;\;=\;\bf{5x}}$

$\\\;\sf{:\mapsto\;\;\;Present\;age\;of\;Heena\;\;=\;\bf{5(3)}}$

$\\\;\sf{:\mapsto\;\;\;Present\;age\;of\;Heena\;\;=\;\bf{15\;\;years}}$

$\\\;\large{\underline{\underline{\rm{Present\;age\;of\;Heena\;is\;\;\boxed{\bf{15\;\;years}}}}}}$

$\\\;\sf{:\mapsto\;\;\;Present\;age\;of\;Meena\;\;=\;\bf{3x}}$

$\\\;\sf{:\mapsto\;\;\;Present\;age\;of\;Meena\;\;=\;\bf{3(3)}}$

$\\\;\sf{:\mapsto\;\;\;Present\;age\;of\;Meena\;\;=\;\bf{9\;\;years}}$

$\\\;\large{\underline{\underline{\rm{Present\;age\;of\;Meena\;is\;\;\boxed{\bf{9\;\;years}}}}}}$

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★ Let's verify our answer :-

For verification, we need to simply apply the values we got into the equation we formed. Then,

$\\\;\tt{\leadsto\;\;\;Present\;\;ratio\;\;of\;\;ages\;\;of\;\;Heena\;\;and\;\;Meena\;\;=\;\;\dfrac{5x}{3x}}$

$\\\;\tt{\leadsto\;\;\;Present\;\;ratio\;\;of\;ages\;\;of\;\;Heena\;\;and\;\;Meena\;=\;\;\dfrac{5(3)}{3(3)}}$

$\\\;\tt{\leadsto\;\;\;Present\;\;ratio\;\;of\;ages\;\;of\;\;Heena\;\;and\;\;Meena\;=\;\;\dfrac{15}{9}}$

Dividing numerator and denominator by 3, we get,

$\\\;\tt{\leadsto\;\;\;Present\;\;ratio\;\;of\;\;ages\;\;of\;\;Heena\;\;and\;\;Meena\;=\;\;\dfrac{5}{3}}$

Clearly, this satisfies the condition of question.

Then, after six years,

$\\\;\;\;\;\;\tt{\leadsto\;\;\;\dfrac{(5x\;\;+\;\;6)}{(3x\;\;+\;\;6)}\;=\;\;\dfrac{7}{5}}$

$\\\;\;\;\;\;\tt{\leadsto\;\;\;\dfrac{(5(3)\;\;+\;\;6)}{(3(3)\;\;+\;\;6)}\;=\;\;\dfrac{7}{5}}$

$\\\;\;\;\;\;\sf{\leadsto\;\;\;\dfrac{(15\;\;+\;\;6)}{(9\;\;+\;\;6)}\;=\;\;\dfrac{7}{5}}$

$\\\;\;\;\;\;\tt{\leadsto\;\;\;\dfrac{21}{15}\;\;=\;\;\dfrac{7}{5}}$

Dividing the numerator and denominator by 3, we get,

$\\\;\;\;\;\;\bf{\leadsto\;\;\;\dfrac{7}{5}\;=\;\dfrac{7}{5}}$

Clearly, LHS = RHS.

Here both the conditions, satisfy. So our answer is correct.

Hence, Verified.

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Answer 2

$\large{\boxed{\boxed{\underbrace{\sf{Question}}}}}$

The ages of Heena and Meena are in the ratio 5:3. After 6 years, their ages will be in the ratio 7:5. Find their present ages.

$\sf Given \: that \begin{cases} & \sf{Heena \: and \: Meena \: age \: in \: ratio = \bf{5:3}} \\ & \sf{After \: 6 \: years \: age \: ratio = \bf{7:5}} \end{cases}$

$\sf To \: find \begin{cases} & \sf{Their \: present \: age} \end{cases}$

$\sf Solution \begin{cases} & \sf{Heena \: present \: age = \bf{15 \: years}} \\ & \sf{Meena \: present \: age = \bf{9 \: years}} \end{cases}$

$\large{\boxed{\boxed{\underbrace{\sf{Understanding \: the \: concept}}}}}$

This question says that tere are two girls namely Heena and Meena. Afterwards it says that we have to find their present age and this is given that Heena and Meena age is in a ratio of 5:3 respectively. Afterwards it says that after 6 years their age come in ratio of 7:5 And (wow) we already know that this question is related to very interesting chapter of mathematics that is Linear equations and variables.

$\large{\boxed{\boxed{\underbrace{\sf{Some \: procedure \: of \: this \: questy}}}}}$

To solve this question we have to use our taken assumptions afterthat finding the ratio of their present age. Afterwards Finding the ratio after 6 years afterthat putting the values. Afterwards we get the value of our taken assumption a . Afterthat substituting the value of a in the present age of Heena. Afterthat substituting the value of a in the present age of Meena.

$\large{\boxed{\boxed{\underbrace{\sf{Using \: concept}}}}}$

✠ $\Large{\boxed{\bf{\frac{5a + 6}{3a + 6} = \frac{7}{5}}}}$

$\large{\boxed{\boxed{\underbrace{\sf{Assumptions}}}}}$

Let, the constant will be a by which both the numerators and the denominators of that initial ratio should be multiplied.

$\large{\boxed{\boxed{\underbrace{\sf{Full \: solution}}}}}$

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✺ The present ratio of their age is 5:3 (Given)

✺ The ratio of their age after 6 years is 7:5 (Given)

✺ The present age of Heena and Meena both (To find)

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As we know that we let, the constant will be a by which both the numerators and the denominators of that initial ratio should be multiplied.

So, continuing....

As we know that the present age of Heena is 5a year's

And the present age of Meena is 3a year's

Hence, the ratio will be $\large{\bf{\frac{5a}{3a}}}$

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As we know that their age ratios will be like 7:5 after six years.

Hence, it will be written as $\large{\boxed{\bf{\frac{5a + 6}{3a + 6} = \frac{7}{5}}}}$

Now, finding the value of a

$\large{\boxed{\bf{\frac{5a + 6}{3a + 6} = \frac{7}{5}}}}$

By cross multiplying the digits we get,

➝ 5(5a+6) = 7(3a+6)

➝ 25a + 30 = 21a + 42

➝ 25a - 21a = 42 - 30

➝ 4a = 12

➝ a = 12/4

➝ a = 3

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Now, finding the present age of Heena first by using the value of a !

➝ Present age of Heena = 5a

➝ Present age of Heena = 5(3)

➝ Present age of Heena = 15

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Now, finding the present age of Meena first by using the value of a !

➝ Present age of Meena = 3a

➝ Present age of Meena = 3(3)

➝ Present age of Meena = 9

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$\small{\boxed{\underline{\sf{Are \: you \: confused \: ? \: \: \: Yes,.}}}}$

$\small{\boxed{\underline{\sf{No, \: need \: to \: worry \: let's \: verify \: it}}}}$

$\small{\boxed{\underline{\underbrace{\sf{Verification}}}}}$

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Present age :

Present age ratio of Heena and Meena age = $\large{\bf{\frac{5a}{3a}}}$

Present age ratio of Heena and Meena age = $\large{\bf{\frac{5(3)}{3(3)}}}$

Present age ratio of Heena and Meena age = $\large{\bf{\frac{15}{9}}}$

Present age ratio of Heena and Meena age = $\large{\bf{\frac{5}{3}}}$

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Age (6) year's :

Age ratio of Heena and Meena age after 6 years = $\large{\bf{\frac{5a + 6}{3a + 6} = \frac{7}{5}}}$

Age ratio of Heena and Meena age after 6 years = $\large{\bf{\frac{5(3) + 6}{3(3) + 6} = \frac{7}{5}}}$

Age ratio of Heena and Meena age after 6 years = $\large{\bf{\frac{15 + 6}{9 + 6} = \frac{7}{5}}}$

Age ratio of Heena and Meena age after 6 years = $\large{\bf{\frac{21}{15} = \frac{7}{5}}}$

Age ratio of Heena and Meena age after 6 years = $\large{\bf{\frac{7}{5} = \frac{7}{5}}}$

Age ratio of Heena and Meena age after 6 years = $\large{\bf{R.H.S = L.H.S}}$

Hence, verified.

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