Question

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

Answer 1

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 $ .