Question

Find which of the following are in proportion: $ 18:9::10:5 $

Answer 1

Hint: Ratio can be expressed as the comparison between two numbers without any units.
Whereas, when the given two ratios are set equal to each other are known as the proportion. Four numbers a, b, c, and d are known to be in the proportion. If $ a:b = c:d $ whereas, four numbers are known as to be in the continued proportion if the terms are expressed as $ a:b = b:c = c:d $ here we will frame the given expression in the mathematical form and will simplify to check the values on both the sides of the equation.

Complete step by step solution:
Given expression: $ 18:9::10:5 $
The above expression can be re-written as –
 $ \dfrac{{18}}{9} = \dfrac{{10}}{5} $
Cross multiply for the above expression, where the numerator of one side is multiplied with the denominator of the opposite side and vice-versa.
 $ 18 \times 5 = 9 \times 10 $
Simplify the above expression finding the product of terms on both sides of the equation.
 $ 90 = 90 $
Both the sides of the above equation are equal hence the given ratios are in the proportion.

Note: The above expression can be solved by the alternative method.
 $ 18:9::10:5 $
The above expression can be re-written as –
 $ \dfrac{{18}}{9} = \dfrac{{10}}{5} $
Find factors of the terms in the numerator on both sides of the equation.
 $ \dfrac{{9 \times 2}}{9} = \dfrac{{5 \times 2}}{5} $
Common factors from the numerator and the denominator cancels each other and therefore remove common factors from the numerator and the denominator on the left hand side of the equation and remove from the numerator and the denominator on the right hand side of the equation
 $ 2 = 2 $
Both the sides of the equation are equal so the given expressions are in the proportion.