Question

If we have 2A=3B and 4B=5C then $ A:C=? $ \[\]
A. $ 4:3 $ \[\]
B. $ 8:15 $ \[\]
C. $ 3:15 $ \[\]
D. $ 15:8 $ \[\]

Answer 1

Hint: We recall the definition of ratio and property of ratio. We divide both sides of $ 2A=3B $ by 3 to express $ A $ in terms of $ B $ . We divide $ 4B=5C $ to express $ C $ in terms of $ B $ . We express the ratio of $ A: C=\dfrac{A}{C} $ and put $ A, C $ in terms of $ B $ to simplify. \[\]

Complete step by step answer:
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 the ratio will not change. It means
\[\begin{align}
  & a:b=ka:kb \\
 & a:b=\dfrac{k}{a}:\dfrac{k}{b} \\
\end{align}\]
  If $ a,b $ are positive integers and they are co-prime then the ratio $ a:b $ is said to be in simplest form. If $ a,b $ are not co-primes then we can divide both $ a $ and $ b $ by the greatest common divisor (gcd) of $ a $ and $ b $ to convert the ratio into the simplest form. \[\]

We are given $ 2A=3B $ and $ 4B=5C $ . We see that in both the equations $ B $ is common. So let us divide both sides of $ 2A=3B $ by 2 and express $ A $ in terms of $ B $ . We have
\[\begin{align}
  & 2A=3B \\
 & \Rightarrow \dfrac{2A}{2}=\dfrac{3B}{2} \\
 & \Rightarrow A=\dfrac{3B}{2} \\
\end{align}\]
So let us divide both sides of $ 4B=5C $ by 5 and express $ C $ in terms of $ B $ . We have
\[\begin{align}
  & 4B=5C \\
 & \Rightarrow \dfrac{4B}{5}=\dfrac{5C}{5} \\
 & \Rightarrow C=\dfrac{4B}{5} \\
\end{align}\]
 We now find the ratio $ A:C $ as;
\[A:C=\dfrac{A}{C}\]
We put the expressions of $ A,C $ in terms of $ B $ and have;
\[\Rightarrow A:C=\dfrac{\dfrac{3}{2}B}{\dfrac{4}{5}B}\]
We divide both numerator and denominator by $ B $ to have;
\[\Rightarrow A:C=\dfrac{\dfrac{3}{2}}{\dfrac{4}{5}}\]
We know how to divide fractions. We multiply the reciprocal of the fraction in the denominator to have;
\[\begin{align}
  & \Rightarrow A:C=\dfrac{3}{2}\times \dfrac{5}{4} \\
 & \Rightarrow A:C=\dfrac{3\times 5}{2\times 4} \\
 & \Rightarrow A:C=\dfrac{15}{8} \\
 & \Rightarrow A:C=15:8 \\
\end{align}\]
So the correct option is D. \[\]

Note:
We note that if $ \dfrac{a}{b} $ then its reciprocal is given as $ \dfrac{b}{a} $ where $ a,b $ cannot be zero. We should always multiply fraction numerator by numerator and denominator by denominator. We note that ratio is used to compare similar quantities and in some units. It cannot be used to compare dissimilar quantities. We should remember that the ratio does not have any unit and is always expressed in the simplest form.