Question

The dinometer 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 3/2. Find the rational number

Answer 1

Answer:

Step-by-step explanation:

f the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is 3/2. Find the rational number. Let the numerator of the rational number be x. So as per the given condition, the denominator will be x + 8.

Answer 2

Given:

• The denominator of a rational number is greater than its numerator by 8.
• And if 17 is added to its numerator, the resultant numerator of the new rational number be 3 and if 1 is subtracted from the denominator, the denominator of the new rational number will be 2.

To find:

• The original rational number.

Solution:

Let the numerator of the rational number be 'f', so the denominator will be (f+8), i.e,

$\: \: \underline{ \boxed{ \sf{ \dfrac{f}{f + 8} }}}\: \: - - - - - - \: \sf{(exp. \: 1)}$

According to the question, if the numerator is increased by 17 and denominator is decreased by 1, we get the expression:

$\: \: \: \underline{ \boxed{ \sf{ \dfrac{f + 17}{(f + 8) - 1} }}}\: \: - - - - - - \: \sf{(exp. \: 2)}$

And this expression equals 3/2, i.e,

$\: \: \underline{ \boxed{ \sf{ \dfrac{f + 17}{(f + 8) - 1} = \dfrac{3}{2} }}}$

Now, to find the the original number, first we've to find the value of 'f', and it can be done by solving the above equation. So, let's start finding it!

$\\ \longmapsto{ \sf{\dfrac{f + 17}{(f + 8) - 1} = \dfrac{3}{2} }} \\ \\ \longmapsto{ \sf{2[f + 17] = 3[(f + 8 )- 1] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bigg( \because Cross - multiplication \bigg)}} \\ \\ \longmapsto{ \sf{(2f) +(2 \times 17) = (3f) + (3 \times 8) - (3 \times 1)}} \\ \\ \longmapsto{ \sf{2f + 34 = 3f + (24 - 3)}} \\ \\ \longmapsto{ \sf{2f + 34 = 3f + 21}} \\ \\ \longmapsto{ \sf{(2f - 3f) = (21 - 34)}} \\ \\ \longmapsto{ \sf{ - 1f = - 13}} \\ \\ \longmapsto{ \sf{f = \dfrac{ \cancel{ - 13}}{ \cancel{- 1}} }} \\ \\ \longmapsto{ \boxed{ \sf{ \red{f = \frak{13}}}}}$

$\:$

We've obtained the value of 'f'. Now, substitute this value in places of 'f' in (exp. 1). The resultant rational number is the original number.

$\longmapsto{ \sf{ \dfrac{f}{f + 8} }} \\ \\ \longmapsto{ \sf{ \dfrac{13}{13 + 8}}} \\ \\ \longmapsto {\boxed{\boxed{ \frak { \red{ \dfrac{13}{21}}}}}}$

Verification:

How to verify? Let's consider the L.H.S as 2/3 and the R.H.S as (exp. 2)! In R.H.S, substitute the values in their respective places and check whether it equals the L.H.S, i.e, 2/3.

$\\ \longmapsto{\sf{ \dfrac{f + 17}{(f + 8) - 1} }} \\ \\ \longmapsto{ \sf{ \dfrac{13 + 17}{(13 + 8) - 1}}} \\ \\ \longmapsto{ \sf{ \dfrac{30}{21 - 1}}} \\ \\ \longmapsto{ \sf{ \dfrac{3 \cancel{0}}{2 \cancel{0}} }} \\ \\ \longmapsto{ \boxed{ \sf{ \dfrac{3}{2} }}}$

L.H.S = R.HS, hence the value of 'f' is correct, so the value of original number is also correct!

$\therefore \underline{ \sf{The \: original \: number \: is \: {\textsf{\textbf{\pink{ \dfrac{13}{21} } \sf{.}}}}}}$

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