question_answer

Determine if the following are in proportion:

(a) 15, 45, 40, 120

(b) 33, 121, 9, 96

(c) 24, 28, 36, 48

(d) 32, 48, 70,210

(e) 4, 6, 8, 12

(f) 33, 44, 75,100

Answer:

(a) We have, \[15:45=\frac{15}{45}=\frac{1}{3}\]

and                   \[40:120=\frac{40}{120}=\frac{1}{3}\]

Since,                \[15:45=40:120\]

So, 15, 45, 40, 120 are in proportion. Other method: We have,

Product of extremes \[=15\times 120=1800\]

Product of middle terms \[=45\times 40=1800\]

Product of extremes = Product of middle terms Hence, 15, 45, 40, 120 are in proportion.

(b) We have \[33:121=\frac{33}{121}=\frac{3}{11}\]

and                      \[9:96=\frac{9}{96}=\frac{3}{32}\]

Therefore,   \[\frac{3}{11}\ne \frac{3}{32}\]

So,            \[~33:121\ne 9:96\]

Thus, 33, 121, 9, 96 are not in proportion. Other method: We have,

Product of extremes \[=33\times 96=3168\]

Product of middle terms \[=121\times 9=1089\]

Product of extremes \[\ne \] Product of middle terms

Hence, 33, 121, 9, 96 are not in proportion.

(c) We have, \[~24:28=\frac{24}{28}=\frac{6}{7}\]

and                    \[36:48=\frac{36}{48}=\frac{3}{4}\]

Since,                \[\frac{6}{7}\ne \frac{3}{4}\]

Therefore,   \[24:28\ne 36:48\]

They, 24, 28, 36, 48 are not in proportion.

(d) We have,     \[32:48=\frac{32}{48}=\frac{2}{3}\]

and                      \[70:210=\frac{70}{210}\text{=}\frac{1}{3}\text{ }\]

Since,                \[\frac{2}{3}\ne \frac{1}{3}\]

Therefore,   \[32:48\ne 70:210\]

Hence, 32, 48, 70, 210 are not in proportion.

(e) We have,      \[4:6=\frac{4}{6}=\frac{2}{3}\]

and                   \[8:12=\frac{8}{12}=\frac{2}{3}\]

So,                   \[4:6=8:12\]

Therefore, 4, 6, 8, 12 are in proportion.

Other method : We have

Product of extremes \[=4\times 12=48\]

Product of middle terms \[=6\times 8=48\]

Product of extremes = product of middle terms Hence, 4, 6, 8, 12 are in proportion.

(f) We have,      \[33:44=\frac{33}{44}=\frac{3}{4}\]

and                   \[75:100=\frac{75}{100}=\frac{3}{4}\]

So,                   \[33:44=75:100\]

Therefore, 33, 44, 75, 100 are in proportion.