Question

36. Two numbers are in the ratio 21:17. If their HCF is 5. find the numbers.​

Answer 1

$\large\sf\underline{Given\::}$

• Ratio of two numbers as 21 : 17

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• Highest Common Factor ( HCF ) is 5

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$\large\sf\underline{To\:find\::}$

• The numbers whose HCF is 5 .

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$\large\sf\underline{Creating\:a\:road\:map\:for\:Solution:}$

Here as we are given the ratio of the number and their HCF and we are asked to find the actual number.

So will use one important property which follows as :

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✪ $\tt\red{LCM × HCF = Product\:of\:two\:numbers}$ .

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In order to use this property we do need the Lowest Common Multiple ( LCM ) of those two numbers. So we will first assume the numbers then find their LCM and proceed with the simple calculation. Let's begin !

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$\large\sf\underline{Assumption\::}$

Let the two numbers be :

• First number = 21x

• Second number = 17x

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$\large\sf\underline{Solution\::}$

We know,

★ $\small\fbox\orange{LCM × HCF = Product\:of\:two\:numbers}$

Now LCM and HCF of those two numbers :

• LCM ( 21x , 17x ) = ${\sf{{\pink{21 \times 17 \times x}}}}$ .

• HCF ( 21x , 17x ) = ${\sf{{\pink{5}}}}$ [ given ] .

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Let's substitute the values in the property :

$\sf\longmapsto\:21 \times 17 \times x \times 5 = 21x \times 17x$

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• Multiplying the terms now

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$\sf\longmapsto\:357 \times x \times 5 = 357x^{2}$

$\sf\longmapsto\:1785x = 357x^{2}$

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• Transposing 357 from RHS to LHS

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$\sf\longmapsto\:\frac{\cancel{1785}x}{\cancel{357}} = x^{2}$

$\sf\longmapsto\:5x = x^{2}$

$\sf\:oR\:x^{2}=5x$

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• Transposing x from RHS to LHS

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$\sf\longmapsto\:\frac{x\cancel{^{2}}}{\cancel{x}}=5$

$\small{\underline{\boxed{\mathrm\pink{\longmapsto\: x\:=\:5}}}}$

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So at last let's substitute the value of x in the assumed value :

• First number = 21x = 21 × 5 = ${\sf{{\purple{105}}}}$

• Second number = 17x = 17 × 5 = ${\sf{{\purple{85}}}}$

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$\dag\:\underline{\sf So\:the\:required\:two\:numbers\:are\:105\:and\:85.}$

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$\large\sf\underline{Verifying\::}$

According to the question :

HCF of two numbers = 5

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So let's see if the HCF of 105 and 85 is 5 or not

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• 105 = 5 × 7 × 3 ( factors of 105 )

• 85 = 5 × 17 ( factors of 85 )

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So their HCF is 5 only . Hence our answer is correct .

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!! Hope it helps !!