CAT 2025 Lesson : Percentages - Of Concept & The Denominator

What percent is 2100 grams of $70$ kg?

Solution

To get an equation, we rewrite the above question as a statement, where the units are the same.

As 1 kg = 1000 grams, the statement here is '$2.1$ kg is ____$\%$ of $70$ kg'.

In this statement, is signifies equal to and of signifies multiplication. Writing this as equation,

$2.1 = x\%$ of $70$

⇒ $2.1 = \dfrac{x}{100} \times 70$

⇒ $x = 3$

$\therefore$ $2.1$kg is 3% of $70$kg.

Alternatively

The question simply means if 70 kg were 100%, then what is 2.1 kg in percentage terms.

$x = \dfrac{2.1}{70} \times$ 100% = 3%

Answer: $3\%$

After working for $24$ hours on a project, Elda realised that $25\%$ of the work still remained. How many more hours will she require to complete the work?

Solution

This question can be solved only if we assume that Elda works at a constant rate. CAT and other tests do not test us with riddles. Therefore, you need to make such assumptions, unless explicitly stated otherwise.

Portion of work completed in $24$ hours = $100\% - 25\% = 75\%$

As Elda works at a constant rate, the statement here is, '$24$ hours is $75\%$ of ____ hours'

$24 = \dfrac{75}{100} \times x$

$x = 32$

$\therefore$ Additional hours taken by Elda = $32 - 24 =$ 8 hours

Answer: $8$ hours

Express $2$ hours $40$ minutes as the percentage of $1$ hour $36$ minutes.

Solution

$2$ hours $40$ minutes = $2 \times 60 + 40 = 160$ minutes

$1$ hour $36$ minutes = $1\times 60 + 36 = 96$ minutes

The statement here is '$160$ minutes is ___$\%$ of $96$ minutes'

The answer is $\dfrac{160}{96} \times 100\% = \dfrac{1000}{6}\% = 166.67\%$

Answer: $166.67\%$

Express $2$ hours $40$ minutes as the percentage of $1$ hour $36$ minutes.

Solution

$2$ hours $40$ minutes = $2 \times 60 + 40 = 160$ minutes

$1$ hour $36$ minutes = $1\times 60 + 36 = 96$ minutes

The statement here is '$160$ minutes is ___$\%$ of $96$ minutes'

The answer is $\dfrac{160}{96} \times 100\% = \dfrac{1000}{6}\% = 166.67\%$

Answer: $166.67\%$

In a farm, wheat production in $2018$ was $150$ tonnes and $2019$ was $165$ tonnes. By what percentage did the wheat production increase in $2019$?

Solution

Increase was $15$ tonnes over the $2018$ production of $150$ tonnes. Therefore, it is implied and understood that the percentage increase is asked relative to the production in $2018$.

Note that it does not make sense to look at an increase in relation to the final value. In this question, it is also implied that we are required to find the annual percentage increase.

$\%$ increase in $2019$ = $\dfrac{15}{150} \times 100\% = 10\%$

Answer: $10\%$