Question

A 24-inch-tall picture is 20% as tall as the ceiling is high. How high is the ceiling?

Answer 1

Hint: A twenty-four-inch-tall picture is equal to twenty percent of the height of a ceiling. Or we say that the twenty percent of height of a ceiling is equal to the product of twenty percent and height of the ceiling which is also equivalent to the product of the decimal zero point two and height of the ceiling.

Complete step-by-step solution:
Consider a ceiling of height \[x\] inches.
Then, twenty percent of the height of the ceiling is equal to the product of twenty percent and height of a ceiling. That is \[20\% \times x\].
Percent sign represents the fraction \[\dfrac{1}{{100}}\], therefore twenty percent means twenty divide by hundred.
\[20\% \equiv 20 \times \dfrac{1}{{100}}\]
\[20\% \equiv \dfrac{{20}}{{100}}\]
Simplify the fraction where twenty as numerator and hundred in denominator as shown below.
\[\dfrac{{20}}{{100}} = \dfrac{2}{{10}}\]
Convert it in decimal form as it is more convenient to multiply with other numbers in this case.
\[\dfrac{2}{{10}} = 0.2\]
Therefore, twenty percent is equivalent to zero point two in decimal form.
\[20\% = 0.2\]
Now, the twenty percent of height of a ceiling which is equal to the product of twenty percent and height of the ceiling is also equivalent to the product of the decimal zero point two and height of the ceiling.
\[20\% \times x = 0.2 \times x\]
According to the question, there is a picture of height twenty-four inches which is equal to the twenty percent of the height of a ceiling.
Therefore, twenty percent of the height of a ceiling (\[0.2 \times x\] inches) is equal to the height of a picture (\[24\] inches).
\[0.2 \times x = 24\]
Solve the equation \[0.2 \times x = 24\] and obtain the value of \[x\] which is nothing but the height of a ceiling as shown below.
Multiply both sides by ten and then divide by two as,
\[ \Rightarrow 10 \times 0.2 \times x = 10 \times 24\]
\[ \Rightarrow 2 \times x = 240\]
\[ \Rightarrow \dfrac{{2 \times x}}{2} = \dfrac{{240}}{2}\]
\[ \Rightarrow x = 120\]
Thus, the height of a ceiling is one hundred twenty inches.

Note: Percentage is a term used for comparison between two quantities, where we assume the quantity of one thing equivalent to hundred and compare others with respect to hundred. Therefore, in this question if we assume ceiling length is hundred then the picture length is twenty it means 20% of the length of ceiling is equal to length of the picture.