Question

If $p:q::r:s$, then the correct statement is ___________
A) $qr = ps$
B) $pqr = s$
C) $qrs = p$
D) $pq = rs$

Answer 1

Hint- The symbol $a:b$ means $\dfrac{a}{b}$ and $a:b::c:d$ means$\dfrac{a}{b} = \dfrac{c}{d}$.

Complete step by step answer:
The given proportion is $p:q::r:s$
Using the given hint we rewrite the given proportion, therefore we get,
 $\dfrac{p}{q} = \dfrac{r}{s}$
Let us cross multiply the above proportion.
After cross multiplication, we obtain $ps = qr$
Changing sides the above expression reduces to $ - qr = - ps$
Cancelling negative signs from both sides, we get, $qr = ps$.
Hence the correct statement is A. $qr = ps$

Additional information: Ratio and proportion plays a significant role in mathematics. In our daily life activities we frequently use the concept of ratio and proportion whether we deal with money or cook a dish in our kitchen. Ratio defines the relation between two quantities $a$ and $b$ such as $a:b$ where $b$ is not equal to zero. It is also written$a:b = \dfrac{a}{b}$ . Thus the ratio of 5 to 30 is represented as$5:30 = 1:6$. Proportion is an equation which gives that the two ratios are equivalent. In the proportion $a:b = 5:20$ the relation of $a$ to $b$ and relation of 5 to 20 are the same. Here 5:20 is a ratio and its value is $\dfrac{1}{4}$ . So the value of $\dfrac{a}{b}$ is also $\dfrac{1}{4}$.

Note: The ratio exists between the same kinds of quantities. When we compare two things the unit must be the same. If a bus runs 40 km/hour and it covers a distance 200 km in 5 hours then in the proportion we have $\dfrac{{40{\text{ km}}}}{{1{\text{ hour}}}} = \dfrac{{200{\text{ km}}}}{{5{\text{ hour}}}}$ . Ratio and proportion are the two faces of the same coin. When two ratios are equal they are said to be in proportion.