Question

Make set of all possible proportions from the numbers 15, 18, 35 and 42.
(a) 15:18::35:42
(b) 15:35::18:42
(c) 42:15=35:18
(d) 42:35=18:15

Answer 1

Hint: To solve this question, we will first break all the four numbers in multiples of primes, then using the definition of proportionality, we will solve each and every option and will see which option satisfies the condition of proportionality. After that, we will make a set of all possible proportions from the numbers given in the question.

Complete step by step answer:
Before we solve this question, let us see what makes two pairs of numbers in proportion.
Proportion says that two ratios or fractions are equal Suppose, we have two pairs of numbers, say a, b and c, d, then we can say that ( a, b ) and ( c, d ) are in proportion if $\dfrac{a}{b}=\dfrac{c}{d}$.
For example, $\dfrac{1}{2}=\dfrac{3}{6}$, as on solving $\dfrac{3}{6}$, we get $\dfrac{1}{2}$.

Now, the number we have is 15, 18, 35, and 42.
So, firstly we factorize the number.
Then, $15=3\times 5$ , $18=3\times 3\times 2$, $35=7\times 5$ and $42=7\times 3\times 2$.
Now, 15 : 18 :: 35 : 42, means $\dfrac{15}{18}::\dfrac{35}{42}$.
We can write $\dfrac{15}{18}::\dfrac{35}{42}$ as $\dfrac{5\times 3}{3\times 3\times 2}::\dfrac{7\times 5}{7\times 3\times 2}$
On solving, we get
$\dfrac{5}{3\times 2}::\dfrac{5}{3\times 2}$
$\dfrac{5}{6}::\dfrac{5}{6}$
Hence, option ( a ) is true.
Now, 15 : 35 :: 18 : 42, means $\dfrac{15}{35}::\dfrac{18}{42}$.
We can write $\dfrac{15}{35}::\dfrac{18}{42}$ as $\dfrac{5\times 3}{7\times 5}::\dfrac{3\times 3\times 2}{7\times 3\times 2}$
On solving, we get
\[\dfrac{3}{7}::\dfrac{3}{7}\]
Hence, option ( b ) is true.
Now, 42 : 15 = 35 : 18, means $\dfrac{42}{15}=\dfrac{35}{18}$.
We can write $\dfrac{42}{15}=\dfrac{35}{18}$ as \[\dfrac{7\times 3\times 2}{5\times 3}=\dfrac{7\times 5}{3\times 3\times 2}\]
On solving, we get
\[\dfrac{7\times 2}{5}=\dfrac{7\times 5}{3\times 3\times 2}\]
\[\dfrac{14}{5}\ne \dfrac{35}{18}\]
Hence, option ( c ) is not true.
Now, 42:35=18:15, means $\dfrac{42}{35}=\dfrac{18}{15}$.
We can write $\dfrac{42}{35}=\dfrac{18}{15}$ as \[\dfrac{7\times 3\times 2}{7\times 5}=\dfrac{3\times 3\times 2}{5\times 3}\]
On solving, we get
\[\dfrac{3\times 2}{5}=\dfrac{3\times 2}{5}\]
\[\dfrac{6}{5}=\dfrac{6}{5}\]
Hence, option ( d ) is true.
So, set of possible proportion from the numbers 15, 18, 35 and 42 is = { 15:18 :: 35:42, 15:35 :: 18:42, 42:35 = 18:15 }

Note:
To solve such questions, one must know the meaning of proportionality, which means when two fractions are equal then we say fraction are in proportion. Also, we have to check each and every option so you must solve questions quickly but carefully too. Try not to make any calculation error.