Question

Six years ago the ratio of ages of Bob and Joe is 2:5. Four years from now the ratio of their ages will be 4:5. Find the sum of their present ages?

 Choose one of the following answers

1

2

5

17

19​

Answer 1

Answer:

19

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Step-by-step explanation:

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Six years ago:

• Bob's age: 2x.

• Joe's age: 5x.

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Their present ages will be:

• Bob's age: 6 + 2x.

• Joe's age: 6 + 5x.

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Four years from now;

• Bob's age: 4x.

• Joe's age: 5x.

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We can say that;

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$\sf \dashrightarrow \ \dfrac{Bob's \ Present \ Age + 4}{Joe's \ Present \ Age + 4} = \dfrac{4}{5}$

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$\sf \dashrightarrow \ \dfrac{6 + 2x + 4}{6 + 5x + 4} = \dfrac{4}{5}$

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$\sf \dashrightarrow \ \dfrac{10 + 2x}{10 + 5x} = \dfrac{4}{5}$

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On cross-multiplying we get;

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$\sf \dashrightarrow \ 5\big\{10 + 2x\big\} = 4\big\{10 + 5x\big\}$

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$\sf \dashrightarrow \ 50 + 10x = 40 + 20x$

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$\sf \dashrightarrow \ 50 - 40 = 20x - 10x$

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$\sf \dashrightarrow \ 10 = 10x$

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$\sf \dashrightarrow \boxed{\sf \ x = 1 \ }$

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On substituting the value of 'x' in the present ages of Bob and Joe we get;

‎‎

Bob's present age;

➝ 6 + 2x

➝ 6 + 2(1)

➝ 8 years old.

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Joe's present age;

➝ 6 + 5x

➝ 6 + 5(1)

➝ 11 years old.

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Now, the sum of both their ages is;

• Sum of their ages = 8 + 11

• Sum of their ages = 19

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Therefore the answer is 19.

Answer 2

Given : Six years ago the ratio of ages of Bob and Joe is 2:5. Four years from now the ratio of their ages will be 4:5.

Exigency to find : The Sum of their present ages .

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

❍ Let's Consider six years ago the ages of Bob and Joe be 2x and 5x yrs , respectively.

Therefore ,

⠀⠀⠀⠀⠀━━ Their Present ages will be :

• Present age of Bob's will be : 2x + 6 yrs .
• Present age of Joe's will be : 5x + 6 yrs .

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:According \: to \: \: the \: Given \: Question \::}}$

━━━━ Four years from now the ratio of their ages will be 4:5 .

$\qquad :\implies \sf \dfrac{ Bob's \:Present \:Age \:+ 4 }{ Joe's \:Present \:Age \:+ 4} = \dfrac{4}{5}$

$\qquad :\implies \sf \dfrac{ 2x + 6 \:+ 4 }{ 5x + 6 \:+ 4} = \dfrac{4}{5}$

$\qquad :\implies \sf \dfrac{ 2x + 10 }{ 5x + 10} = \dfrac{4}{5}$

⠀⠀⠀⠀ By Cross Multiplication :

$\qquad :\implies \sf \dfrac{ 2x + 10 }{ 5x + 10} = \dfrac{4}{5}$

$\qquad :\implies \sf 5( 2x + 10) = 4( 5x + 10)$

$\qquad :\implies \sf 10x + 50 = 4( 5x + 10)$

$\qquad :\implies \sf 10x + 50 = 20x + 40$

$\qquad :\implies \sf 10x + 50 - 40 = 20x$

$\qquad :\implies \sf 10x + 10 = 20x$

$\qquad :\implies \sf 20x - 10x = 10$

$\qquad :\implies \sf 10x = 10$

$\qquad :\implies \sf x = \cancel {\dfrac{10}{10}}$

$\qquad \longmapsto \frak{\underline{\purple{\:x = 1 \:yrs }} }\bigstar$

Therefore,

• Bob's Present age is 2x + 6 = 2(1) + 6 = 2 + 6 = 8 yrs .

• Joe's Present age is 5x + 6 = 5(1) + 6 = 5 + 6 = 11 yrs .

Now ,

⠀⠀⠀⠀⠀━━ Sum of there Present ages :

⠀⠀⠀⠀Sum of there ages = Bob's Age + Joe's Age

⠀⠀⠀⠀Sum of there ages = 11 + 8

⠀⠀⠀⠀Sum of there ages = 19 yrs .

Therefore,

⠀⠀⠀⠀⠀$\therefore {\underline{ \mathrm {\:Sum \:of\:there\:Present \:ages\:is\:\bf{19}}}}$

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