Question

If a number is subtracted from the numerator of the fraction 3/5 and thrice that number is added to the denominator, the fraction becomes 1/4. Find the number.
​

Answer 1

$\huge{\sf\color{green}\underline{Solution}}$

$\sf\color{yellow}\underline{Let \: }$

• The number be x.

$\sf\color{cyan}\underline{A/Q \: }$

$\sf\pink{:\longmapsto \: \: \: \: \: \: \: \: \:{\dfrac{3 - x}{5 + 3x} = \dfrac{1}{4}}} \:$

Cross Multiplying

$\sf{:\longmapsto} \: \: \: \: \: \: \: \: \: {4(3 - x) = 5 + 3x}$

$\sf{:\longmapsto} \: \: \: \: \: \: \: \: \: {12 - 4x = 5 + 3x}$

Transposing 4x to RHS and 5 to LHS

$\sf{:\longmapsto} \: \: \: \: \: \: \: \: \: {12 - 5 = 4x + 3x}$

$\sf{:\longmapsto} \: \: \: \: \: \: \: \: \:{7 = 7x}$

$\sf{:\longmapsto} \: \: \: \: \: \: \: \: \:{7x = 7}$

Dividing

$\sf{:\longmapsto} \: \: \: \: \: \: \: \: \:{x = \cancel\dfrac{7}{7}}$

$\sf\blue{:\longmapsto} \: \: \: \: \: \: \: \: \:{\red{\underline{\boxed{\bf{{x = 1}}}}}}$

$\sf\large{\orange{\therefore{\underline{The \: required \: number \: is \: 1. \: }}}}$

$\red{\rule{200pt}{2pt}}$

$\sf{\large\color{yellow}\underline{Verification}}$

Substituting x = 1 on the LHS, we get

$\sf{\dfrac{3 - 1}{5 + 3(1)}}$

$\sf=\cancel{\dfrac{2}{8}}$

$\sf=\dfrac{1}{4}.$

RHS = $\sf\dfrac{1}{4}.$

LHS = RHS

1/4 = 1/4

Hence, verified! ✓.

Steps for solving linear equations based word problems

1. Read the problem carefully to analyse the facts given.
2. Denote the unknown quantity by x or by any other variable.
3. Express all other quantities mentioned in the problem in terms of the variable.
4. Frame an equation by using the conditions of the problems.
5. Solve the equation to find the value of the variable or the unknown. If the conditions of the given problem are satisfied by the value of the unknown, the solution is correct.

Steps for solving linear equations

1. Simplify both the sides of the equation. Use the distributive law (if necessary) to separate the terms containing the variable from the constant terms.
2. If the equation involves fractions, multiply both the sides by the LCM of the denominators to clear the fractions.
3. If decimals are present, multiply both the sides by a suitable power of 10 to eliminate the decimals.
4. Collect all the terms containing the variable on one side of the equation (generally, the LHS) and all the constant terms on the other side.
5. Divide both the sides of the equation by the resulting coefficient of the variable.

$\space\space\space\orange{\bigstar}{\color{yellow}{\rule{150pt}{4pt}}\orange{\bigstar}}$

Answer 2

Given :

• If a number. is subtracted from the numerator of the fraction 3/5 and thrice thrice the number is added to denominator, the fraction becomes 1/4 .

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

To Find :

• Find the number.

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

Solution :

~ According to the Question :

➳ If a number is subtracted from the numerator of the fraction 3/5 and thrice that number is added to the denominator, the fraction becomes 1/4. Let the no. be y .Hence,

$\large{\qquad{\gray{\bigstar}} \: \: {\underline{\underline{\red{\sf{ \dfrac{3 - y}{5 + 3y} = \color{cyan}{\dfrac{1}{4}} }}}}}}$

$\qquad{━━━━━━━━━━━━━━━━━━━━━━}$

~ Now Cross Multiplication :

${\implies{\qquad{\sf{ \dfrac{3 - y}{5 + 3y} = \dfrac{1}{4}}}}} \\ \\ \ {\implies{\qquad{\sf{ 4(3 - y) = 1(5 + 3y) }}}} \\ \\ \ {\implies{\qquad{\sf{ 12 - 4y = 5 + 3y}}}} \\ \\ \ {\implies{\qquad{\sf{ 12 - 5 = 4y + 3y }}}} \\ \\ \ {\implies{\qquad{\sf{ 7 = 7y}}}} \\ \\ \ {\implies{\qquad{\sf{ y = \cancel\dfrac{7}{7} }}}} \\ \\ \ {\qquad{\sf{ Value \: of \: y \: = {\color{maroon}{\sf{ 1}}}}}}$

$\qquad{━━━━━━━━━━━━━━━━━━━━━━}$

Therefore :

$\large{\red{\dashrightarrow{\orange{\underline{\underline{\green{\sf{ Required \: Number = 1}}}}}}}}$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

Verification :

${\leadsto{\sf{ \dfrac{3 - y}{5 + 3y} = \dfrac{1}{4}}}} \\ \\ {\leadsto{\sf{ \dfrac{3 - 1}{5 + 3 \times 1 } = \dfrac{1}{4}}}} \\ \\ {\leadsto{\sf{ \dfrac{2 }{5 + 3} = \dfrac{1}{4}}}} \\ \\ {\leadsto{\sf{ \cancel\dfrac{2}{8} = \dfrac{1}{4}}}} \\ \\ {\leadsto{\red{\underline{\sf{ \dfrac{1}{4} = \dfrac{1}{4}}}}}} \\ \\ {\red{\underline{\sf{ LHS = RHS}}}}$

Hence, Verified.