Question

Arrange the given fractions in descending order of magnitude:
(i) \[\dfrac{5}{{16}},\dfrac{{13}}{{24}},\dfrac{7}{8}\]
(ii) \[\dfrac{4}{5},\dfrac{7}{{15}},\dfrac{{11}}{{20}},\dfrac{3}{4}\]
(iii) \[\dfrac{5}{7},\dfrac{3}{8},\dfrac{9}{{11}}\]

Answer 1

Hint:First we find the LCM of the denominators of the fractions. Then multiply both the numerator and denominator of each fraction by LCM to make the denominators same. Then by arranging the numerators in descending order after multiplying both the numerator and denominator of each fraction by LCM. After simplifying we get the fractions in descending order.

Complete step by step answer:
(i) Given \[\dfrac{5}{{16}},\dfrac{{13}}{{24}},\dfrac{7}{8}\]----(1)
Then the LCM of \[16,24,8\] is \[48\]
Multiply both the numerator and denominator of each fraction of the expression (1) by \[48\], we get
\[\dfrac{5}{{16}} \times \dfrac{{48}}{{48}},\dfrac{{13}}{{24}} \times \dfrac{{48}}{{48}},\dfrac{7}{8} \times \dfrac{{48}}{{48}}\]
\[ \Rightarrow \]\[\dfrac{{15}}{{48}},\dfrac{{26}}{{48}},\dfrac{{42}}{{48}}\]--(2)
Since the denominator of each fraction of the expression (2) is the same. Hence arranging the numerators in descending order, we get
\[\dfrac{{42}}{{48}},\dfrac{{26}}{{48}},\dfrac{{15}}{{48}}\]---(3)
Simplify the expression (3), we get the given fractions in descending order
i.e., \[\dfrac{7}{8},\dfrac{{13}}{{24}},\dfrac{5}{{16}}\].

(ii) Given \[\dfrac{4}{5},\dfrac{7}{{15}},\dfrac{{11}}{{20}},\dfrac{3}{4}\]----(14)
Then the LCM of \[5,15,20,4\] is \[60\]
Multiply both the numerator and denominator of each fraction of the expression (4) by \[60\], we get
\[\dfrac{4}{5} \times \dfrac{{60}}{{60}},\dfrac{7}{{15}} \times \dfrac{{60}}{{60}},\dfrac{{11}}{{20}} \times \dfrac{{60}}{{60}},\dfrac{3}{4} \times \dfrac{{60}}{{60}}\]
\[ \Rightarrow \]\[\dfrac{{48}}{{60}},\dfrac{{28}}{{60}},\dfrac{{33}}{{60}},\dfrac{{45}}{{60}}\]--(5)
Since the denominator of each fraction of the expression (5) is the same. Hence arranging the numerators in descending order, we get
\[\dfrac{{48}}{{60}},\dfrac{{45}}{{60}},\dfrac{{33}}{{60}},\dfrac{{28}}{{60}}\]---(6)
Simplify the expression (6), we get the given fractions in descending order
i.e., \[\dfrac{4}{5},\dfrac{3}{4},\dfrac{{11}}{{20}},\dfrac{7}{{15}}\].

(iii) Given \[\dfrac{5}{7},\dfrac{3}{8},\dfrac{9}{{11}}\]----(7)
Then the LCM of \[7,8,11\] is \[616\]
Multiply both the numerator and denominator of each fraction of the expression (7) by \[616\], we get
\[\dfrac{5}{7} \times \dfrac{{616}}{{616}},\dfrac{3}{8} \times \dfrac{{616}}{{616}},\dfrac{9}{{11}} \times \dfrac{{616}}{{616}}\]
\[ \Rightarrow \]\[\dfrac{{440}}{{616}},\dfrac{{231}}{{616}},\dfrac{{504}}{{616}}\]--(8)
Since the denominator of each fraction of the expression (8) is the same. Hence arranging the numerators in descending order, we get
\[\dfrac{{504}}{{616}},\dfrac{{440}}{{616}},\dfrac{{231}}{{616}}\]---(9)
Simplify the expression (9), we get the given fractions in descending order
i.e., \[\dfrac{9}{{11}},\dfrac{5}{7},\dfrac{3}{8}\].

Note: To find the LCM of some given integers we use prime factorization. Prime factorization involves breaking down each of the numbers being compared into its product of prime numbers. The LCM is then determined by multiplying the highest power of each prime number together.