Question

State true or false:
\[8:15 > 11:20\].
A) True
B) False

Answer 1

Hint:
A ratio is an expression which is used to compare two similar quantities. In this question we have to compare two given ratios. For this first express the given ratios as fractions. Next compare these two fractions and find which one is greater.

Complete step by step solution: In this question we have to compare two ratios. First convert the given ratios to fractions. Any ratio \[a:b\] can be expressed as a fraction \[\dfrac{a}{b}\]. Therefore \[8:15\] can be expressed as \[\dfrac{8}{{15}}\] and \[11:20\] can be expressed as \[\dfrac{{11}}{{20}}\].
Next compare the two fractions \[\dfrac{8}{{15}}\] and \[\dfrac{{11}}{{20}}\].
We use the common denominator method:
To make the denominator of the two fractions the same, first find the L.C.M. of the two denominators \[15\] and \[20\]. The L.C.M. is \[60\]. So we have to convert the denominators of both the fractions to \[60\] and the numerator will also change accordingly.
\[\dfrac{{8 \times 4}}{{15 \times 4}} = \dfrac{{32}}{{60}}\]
\[\dfrac{{11 \times 3}}{{20 \times 3}} = \dfrac{{33}}{{60}}\]
Hence observe that comparing \[\dfrac{8}{{15}}\] and \[\dfrac{{11}}{{20}}\] is equivalent to comparing \[\dfrac{{32}}{{60}}\] and \[\dfrac{{33}}{{60}}\].
Comparing:
\[\dfrac{{32}}{{60}}\] and \[\dfrac{{33}}{{60}}\], the denominator is same clearly \[33 > 32\] therefore \[\dfrac{{33}}{{60}} > \dfrac{{32}}{{60}}\].
Hence \[\dfrac{{11}}{{20}} > \dfrac{8}{{15}}\]

Therefore the given statement is (b) false.

Note:
Ratios do not have any units. The above comparison of fractions could also be done by using the decimal method. In this the fractions are converted to decimals. \[\dfrac{8}{{15}} = 0.533\] and \[\dfrac{{11}}{{20}} = 0.55\]. Clearly \[0.55 > 0.53\], therefore \[\dfrac{{11}}{{20}} > \dfrac{8}{{15}}\].