# The sum of two numbers is \[\dfrac{{23}}{{20}}\] of the first number. What percentage of the first number is the second number?

(a) 5 %

(b) 10 %

(c) 15 %

(d) 20 %

Answer

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Hint: Start solving by finding the relation between the first number and the second number. Then find the percentage using the percentage formula or obtain the ratio of second number to the first number and multiply by 100.

Let us first assign the numbers to variables.

Let the first number be denoted by variable \[a\] and the second number be denoted by variable \[b\] .

It is given that the sum of these two numbers is \[\dfrac{{23}}{{20}}\] of the first number.

Let us write the equation as follows:

\[a + b = \dfrac{{23}}{{20}}a{\text{ }}..........(1)\]

Now let us simplify this further to obtain the relationship between the two numbers, \[a\] and \[b\] .

In equation (1), taking \[a\]in the left-hand side of the equation to the right-hand side, we get

\[b = \dfrac{{23}}{{20}}a - a\]

Taking \[a\] as a common term and solving, we get:

\[b = \left( {\dfrac{{23}}{{20}} - 1} \right)a\]

\[b = \left( {\dfrac{{23 - 20}}{{20}}} \right)a\]

\[b = \dfrac{3}{{20}}a{\text{ }}..........{\text{(2)}}\]

Hence, we obtained a relation between \[a\] and \[b\] .

To determine what percentage of the first number is the second number, we need to divide the second number by the first number and multiply the result by 100. This comes from the basic percentage formula as follows:

\[{\text{Percentage}} = \dfrac{{{\text{Required Value}}}}{{{\text{Total Value}}}} \times 100{\text{ \% }}\]

Here, the required value is the second number \[b\] and the total value is the first number \[a\] .

The required formula is as follows:

\[{\text{Percentage}} = \dfrac{b}{a} \times 100{\text{ \% }}...........{\text{(3)}}\]

Substituting equation (2) in equation (3), we get:

\[{\text{Percentage}} = \dfrac{{\dfrac{3}{{20}}a}}{a} \times 100{\text{ \% }}\]

Cancelling \[a\] in the numerator and denominator, we get:

\[{\text{Percentage}} = \dfrac{3}{{20}} \times 100{\text{ }}\% \]

Simplifying further we obtain:

\[{\text{Percentage}} = \dfrac{{300}}{{20}}{\text{ \% }}\]

\[{\text{Percentage}} = 15{\text{ \% }}\]

Hence, the correct answer is option (c).

Note: A common mistake committed is writing the equation as \[a + b = \dfrac{{23}}{{20}}b\] with \[b\] being the second number, which is wrong. You can also proceed by finding the ratio \[\dfrac{b}{a}\] directly from the equation \[b = \dfrac{3}{{20}}a{\text{ }}\] and multiplying the result by 100 to get the final answer.

Let us first assign the numbers to variables.

Let the first number be denoted by variable \[a\] and the second number be denoted by variable \[b\] .

It is given that the sum of these two numbers is \[\dfrac{{23}}{{20}}\] of the first number.

Let us write the equation as follows:

\[a + b = \dfrac{{23}}{{20}}a{\text{ }}..........(1)\]

Now let us simplify this further to obtain the relationship between the two numbers, \[a\] and \[b\] .

In equation (1), taking \[a\]in the left-hand side of the equation to the right-hand side, we get

\[b = \dfrac{{23}}{{20}}a - a\]

Taking \[a\] as a common term and solving, we get:

\[b = \left( {\dfrac{{23}}{{20}} - 1} \right)a\]

\[b = \left( {\dfrac{{23 - 20}}{{20}}} \right)a\]

\[b = \dfrac{3}{{20}}a{\text{ }}..........{\text{(2)}}\]

Hence, we obtained a relation between \[a\] and \[b\] .

To determine what percentage of the first number is the second number, we need to divide the second number by the first number and multiply the result by 100. This comes from the basic percentage formula as follows:

\[{\text{Percentage}} = \dfrac{{{\text{Required Value}}}}{{{\text{Total Value}}}} \times 100{\text{ \% }}\]

Here, the required value is the second number \[b\] and the total value is the first number \[a\] .

The required formula is as follows:

\[{\text{Percentage}} = \dfrac{b}{a} \times 100{\text{ \% }}...........{\text{(3)}}\]

Substituting equation (2) in equation (3), we get:

\[{\text{Percentage}} = \dfrac{{\dfrac{3}{{20}}a}}{a} \times 100{\text{ \% }}\]

Cancelling \[a\] in the numerator and denominator, we get:

\[{\text{Percentage}} = \dfrac{3}{{20}} \times 100{\text{ }}\% \]

Simplifying further we obtain:

\[{\text{Percentage}} = \dfrac{{300}}{{20}}{\text{ \% }}\]

\[{\text{Percentage}} = 15{\text{ \% }}\]

Hence, the correct answer is option (c).

Note: A common mistake committed is writing the equation as \[a + b = \dfrac{{23}}{{20}}b\] with \[b\] being the second number, which is wrong. You can also proceed by finding the ratio \[\dfrac{b}{a}\] directly from the equation \[b = \dfrac{3}{{20}}a{\text{ }}\] and multiplying the result by 100 to get the final answer.

Last updated date: 29th Sep 2023

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