Question

the numerator of a fraction is 4 less than the denominator if 1 is add to both it numerator and denominator it's become 1/2 find the fraction.

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Answer 1

Answer:

Step-by-step explanation:

ANSWER:-

Given that:-

• The numerator of fraction is 4 less than its denominator
• 1 is added to both, then,
• It becomes 1/2.

According to Question,

• Let us assume the denominator at x.
• So, the numerator is x-4.

$\sf{\green{\dfrac{x-4+1}{x+1} = \dfrac{1}{2}}}$

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

Cross Multiplication:-

$\sf{2(x-3) = 1(x+1)}$

$\sf{2x-6 = x+1}$

$\sf{2x-x = 1+6}$

$\sf{x = 7}$

• We got the denominator as 7.
• So, numerator will be 7-4 =3

• And fraction is $\sf{\dfrac{3}{7}}$

Verification:-

$\sf{x = 7, Adding \ it}$

$\sf{\green{\dfrac{7-4+1}{7+1} | \dfrac{1}{2}}}$

$\sf{\green{\dfrac{4}{8} | \dfrac{1}{2}}}$

$\sf{\green{\dfrac{1}{2} | \dfrac{1}{2}}}$

Hence verified :D

Note:-

• While attempting this question, do take a constant such as x.
• This is to be done to make equations, make prospective of the question.
• Do not do it 4-x, number is assumed greater, always.
• Down the cliff at @ Idyllic Aurora's answer for 2 variable.

Answer 2

Answer :-

$\: \: \: \underline{\boxed{\sf{\orange{\mapsto \: \: Let's \: see \: to \: what \: the \: question \: says \: !}}}}$

Here we can see that two unknowns values are given which we need to find that is numerator and denominator. How will we find that ? We can use Linear Equation in Two Variables to find it. We can make the value of one variable depend on other to find both. Standard form of Linear Equation in Two Variable is given as :-

• ax + by + c = 0

• px + qy + d = 0

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★ Question :-

The numerator of a fraction is 4 less than the denominator. If 1 is add to both it numerator and denominator it's become 1/2. Find the fraction.

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★ Solution :-

Given,

» The numerator of a fraction = The denominator of fraction - 4

» (1 + The numerator) and (1 + The Denominator) = ½

• Let the numerator of Fraction be 'x'

• Let the denominator of Fraction be 'y'

Then, according to the question,

~ Case I :-

➮ x = y - 4 ... (i)

~ Case II :-

$\: \: \huge{\bf{\purple{\Longrightarrow \: \: \dfrac{x \: + \: 1}{y \: + \: 1} \: = \: \dfrac{1}{2}}}}$

By cross multiplication,

➮ 2(x + 1) = 1(y + 1)

➮ 2x + 2 = y + 1 ... (ii)

From equation (i) and (ii) , we get,

➔ 2(y - 4) + 2 = y + 1

➔ -8 + 2y + 2 = y + 1

➔ 2y - y = 1 + 8 - 2

➔ y = 7

• Hence, the denominator of fraction = y = 7

Now from equation (i) and value of y, we get,

➔ x = y - 4

➔ x = 7 - 4

➔ x = 3

• Hence, the numerator of fraction = x = 3

Now we see that both numerator and denominator have -ve sign. So we can cancel them off.

$\: \: \underline{\boxed{\sf{\blue{\mapsto \: \: \: Hence, \: the \: required \: fraction \: is \: \: \dfrac{3}{7}}}}}$

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$\: \: \underline{\boxed{\rm{\green{\Longrightarrow \: \: \: Confused?, \: Don't \: worry \: let's \: verify \: it}}}}$

Here for verification, we can simply apply the values we got, into the equations we formed. Let's do it !

~ Case I :-

✰ x = y - 4

✰ 3 = 7 - 4

✰ 3 = 3

Clearly, LHS = RHS

~ Case II :-

$\: \: \huge{\bf{\pink{\Longrightarrow \: \: \dfrac{x \: + \: 1}{y \: + \: 1} \: = \: \dfrac{1}{2}}}}$

✰ 2(x + 1) = 1(y + 1)

✰ 2(3 + 1) = 1(7 + 1)

✰ 6 + 2 = 7 + (1)

✰ (8) = (8)

Clearly, LHS = RHS

We see that both the conditions satisfy, so our answer is correct. Hence, Verified.

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$\: \: \underline{\boxed{\sf{Let's \: know \: more \: :-}}}$

• Linear Equations are the equations formed using constant and varible terms in a systematic order, so that on solving them the resultant should come to zero.

• Polynomial are the equation formed using constant and variable term but they can be of any degree unlike Linear Equation which are of only 1 degree that is 1.

*Note :- Here I have solved the question using Two Variable x and y. By this way, its easy to get answer. But the above answer given above by @TheMoonlightPhoenix is using One Variable. Its also correct way. Please refer to it.