Which number should be added to each term of the ratio 7 is to 11 to make it equal to 2 is to 3?

• Aptitude
• Ratio and Proportions

Correct Answer:

Description for Correct answer:
\( \Large \frac{A}{B} = \frac{7}{11} \) (Given)

Let x be added to both A & B

=> \( \Large \frac{7 + x}{11 + x} = \frac{3}{4} \)

Cross multiply the equation

28 + 4x = 33 + 3x

x = 5

Part of solved Ratio and Proportions questions and answers : >> Aptitude >> Ratio and Proportions

Answer

Verified

Hint:To find the term needed to add to \[9:16\] to make \[2:3\], we assume that the terms given are \[x\] in both the denominator and the numerator with the ratio \[9:16\] added to which is equal to \[2:3\] . After placing the values in the ratio we find the value of \[x\] by cross multiplying the ratios.

Complete step by step solution:
Let us assume that the number needed to be added is taken as \[x\].
Now to find the term\ratio needed i.e. \[2:3\] to add the unknown term of \[x\] to the numerator and denominator of the previous ratio of \[9:16\].
The value of the ratio on the LHS of the equation down below is equal to the ratio on the RHS when \[x\] is added in both the numerator and denominator of the LHS given.
\[\Rightarrow \dfrac{9+x}{16+x}=\dfrac{2}{3}\]
Cross multiplying the value of the ratio of the LHS and RHS, we get the value of the unknown variable of \[x\] as:
\[\Rightarrow 3\left( 9+x \right)=2\left( 16+x \right)\]
\[\Rightarrow 27+3x=32+2x\]
\[\Rightarrow x=5\]
Therefore, the value needed to be added with the ratio of \[9:16\] to get \[2:3\] is \[5\].

Note: Student may go wrong if they try to add a single number instead of adding two number in both the denominator and numerator as given below:
\[\Rightarrow \dfrac{9+x}{16+x}=\dfrac{2}{3}\] correct form
\[\Rightarrow \dfrac{9}{16}+x=\dfrac{2}{3}\] Incorrect form

What number must be added to each term of the ratio $ 5:37 $ to make it equal to $ 1:3 $ ? \[\]

Answer

Verified

Hint: We recall the definition and properties of the ratio. We assume the number which needs to be added is $ x $. The ratio now changes to $ 5+x:37+x $ . In accordance with the question, we equate to the ratio $ 1:3 $ and then solve for the unknown $ x $ . \[\]

Complete step by step answer:
We know that a ratio is a fraction with both numerator and denominator as positive numbers. If $ a $ and $ b $ are two positive numbers then the ratio from $ a $ to $ b $ is given as
\[a:b=\dfrac{a}{b}\]
We can multiply or divide a positive number $ k $ and the value of ratio will not change. It means
\[\begin{align}
  & a:b=ka:kb \\
 & a:b=\dfrac{k}{a}:\dfrac{k}{b} \\
\end{align}\]
We are asked in the question to find the number must be added to each term of the ratio $ 5:37 $ to make it equal to $ 1:3 $ . Let u assume the number to be added is $ x $ . So now we have the ratio
\[5+x:37+x=\dfrac{5+x}{37+x}\]
The above ratio is equal to the ratio $ 1:3=\dfrac{1}{3} $ . So we have;
\[\dfrac{5+x}{37+x}=\dfrac{1}{3}\]
 We cross multiply to have;
\[\begin{align}
  & \Rightarrow \left( 5+x \right)\times 3=\left( 37+x \right)\times 1 \\
 & \Rightarrow 5\times 3+x\times 3=37\times 1+x\times 1 \\
 & \Rightarrow 15+3x=37+x \\
\end{align}\]
We subtract 15 both sides to have;
\[\Rightarrow 3x=22+x\]
We subtract $ x $ both sides to have;
\[\Rightarrow 2x=22\]
We divided both sides by 2 to have;
\[\Rightarrow x=11\]
Therefore number 11 should be added to each term of the ratio $ 5:37 $ to make it equal to $ 1:3 $ . \[\]

Note:
We should remember when solving linear equations in one variable we should try collect variable terms at one side and constant terms at the other side of the equation. We note that if $ a,b $ are positive integers and they are co-prime then the ratio $ a:b $ is said to be in the simplest form. I. We use the ratio to compare two same type of quantities which means $ a,b $ must be of the same type and same units. The equality $ a:b=c:d $ is called a proportion that is $ a:b::c:d $ like in this problem we have the proportion $ 5+x:37+x::1:3 $ .

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