Answer:
Step-by-step explanation:
Question (1). An alloy contains zinc and copper in the ratio of 7 : 9.
If the weight of an alloy = x kgs
Then weight of copper = [tex]\frac{9}{7+9}\times (x)[/tex]
= [tex]\frac{9}{16}\times (x)[/tex]
And the weight of zinc = [tex]\frac{7}{7+9}\times (x)[/tex]
= [tex]\frac{7}{16}\times (x)[/tex]
If the weight of zinc = 31.5 kg
31.5 = [tex]\frac{7}{16}\times (x)[/tex]
x = [tex]\frac{16\times 31.5}{7}[/tex]
x = 72 kgs
Therefore, weight of copper = [tex]\frac{9}{16}\times (72)[/tex]
= 40.5 kgs
2). i). 2 : 3 = [tex]\frac{2}{3}[/tex]
4 : 5 = [tex]\frac{4}{5}[/tex]
Now we will equalize the denominators of each fraction to compare the ratios.
[tex]\frac{2}{3}\times \frac{5}{5}[/tex] = [tex]\frac{10}{15}[/tex]
[tex]\frac{4}{5}\times \frac{3}{3}=\frac{12}{15}[/tex]
Since, [tex]\frac{12}{15}>\frac{10}{15}[/tex]
Therefore, 4 : 5 > 2 : 3
ii). 11 : 19 = [tex]\frac{11}{19}[/tex]
19 : 21 = [tex]\frac{19}{21}[/tex]
By equalizing denominators of the given fractions,
[tex]\frac{11}{19}\times \frac{21}{21}=\frac{231}{399}[/tex]
And [tex]\frac{19}{21}\times \frac{19}{19}=\frac{361}{399}[/tex]
Since, [tex]\frac{361}{399}>\frac{231}{399}[/tex]
Therefore, 19 : 21 > 11 : 19
iii). [tex]\frac{1}{2}:\frac{1}{3}=\frac{1}{2}\times \frac{3}{1}[/tex]
[tex]=\frac{3}{2}[/tex]
[tex]\frac{1}{3}:\frac{1}{4}=\frac{1}{3}\times \frac{4}{1}[/tex]
= [tex]\frac{4}{3}[/tex]
Now we equalize the denominators of the fractions,
[tex]\frac{3}{2}\times \frac{3}{3}=\frac{9}{6}[/tex]
And [tex]\frac{4}{3}\times \frac{2}{2}=\frac{8}{6}[/tex]
Since [tex]\frac{9}{6}>\frac{8}{6}[/tex]
Therefore, [tex]\frac{1}{2}:\frac{1}{3}>\frac{1}{3}:\frac{1}{4}[/tex] will be the answer.
IV). [tex]1\frac{1}{5}:1\frac{1}{3}=\frac{6}{5}:\frac{4}{3}[/tex]
[tex]=\frac{6}{5}\times \frac{3}{4}[/tex]
[tex]=\frac{18}{20}[/tex]
[tex]=\frac{9}{10}[/tex]
Similarly, [tex]\frac{2}{5}:\frac{3}{2}=\frac{2}{5}\times \frac{2}{3}[/tex]
[tex]=\frac{4}{15}[/tex]
By equalizing the denominators,
[tex]\frac{9}{10}\times \frac{30}{30}=\frac{270}{300}[/tex]
Similarly, [tex]\frac{4}{15}\times \frac{20}{20}=\frac{80}{300}[/tex]
Since [tex]\frac{270}{300}>\frac{80}{300}[/tex]
Therefore, [tex]1\frac{1}{5}:1\frac{1}{3}>\frac{2}{5}:\frac{3}{2}[/tex]
V). If a : b = 6 : 5
[tex]\frac{a}{b}=\frac{6}{5}[/tex]
[tex]=\frac{6}{5}\times \frac{2}{2}[/tex]
[tex]=\frac{12}{10}[/tex]
And b : c = 10 : 9
[tex]\frac{b}{c}=\frac{10}{9}[/tex]
Since a : b = 12 : 10
And b : c = 10 : 9
Since b = 10 is common in both the ratios,
Therefore, combined form of the ratios will be,
a : b : c = 12 : 10 : 9
