Question

Are \[39\] liters \[:{\text{ }}65\] liters \[::{\text{ }}6\] bottles \[:{\text{ }}10\] bottles are in proportion.

Answer 1

Hint: A ratio is a relationship between two numbers that define the quantity of the first in comparison to the second. For example, For most – mammals, the ratio of legs to noses is \[4:1\], but for humans, the ratio of legs to noses is \[2:1\], Ratios can be written in the fractional form, so comparing \[3\] boys with \[5\] girls could be written 3:5 or $\dfrac{3}{5}$. The order of a ratio may be thought of as “the numerator followed by a denominator“.

Complete step by step answer:
A proportion is simply a statement that two ratios are equal. It can be written in two ways:
as two equal fractions $\dfrac{a}{b}$\[ = \]$\dfrac{c}{d}$; or using a colon, \[a:b = c:d\].

To find the cross products of a proportion, we multiply the outer terms, called the extremes and the middle terms called the means.
New, \[39{\text{ }}litres:65{\text{ }}litres{\text{ }} = \]$\dfrac{{39}}{{65}}$\[[\] cancelling both the terms with \[13]\]
\[ = \]$\dfrac{3}{5}$\[ = 3:5\]
And \[6{\text{ }}bottles:{\text{ }}10{\text{ }}bottles{\text{ }} = \]$\dfrac{6}{{10}}$ \[[\] cancelling both the terms with \[2{\text{ }}]\]
\[ = \]$\dfrac{3}{5}$ \[ = {\text{ }}3:5\]
Since the ratio of both is equal, they are in proportion.

Note: In this type of problem , if \[a:b{\text{ }}::{\text{ }}c:d\] are proportion, then , $\dfrac{a}{b}$ \[ = \] $\dfrac{c}{d}$ or ad \[ = {\text{ }}bc\]
Example\[:\] if $\dfrac{2}{3}$\[ = \]$\dfrac{4}{6}$\[ = \] then,

$2 \times 6$\[ = \]$3 \times 4$
\[{\mathbf{12}}{\text{ }} = {\mathbf{12}}\]
When one of the four numbers in a proportion is unknown, cross products may be used to find the unknown number.