Finding a fraction of a quantity means finding a part of a whole amount.
In many word problems, the word of means multiplication.
1. Main rule
If the fraction is \(a/b\) and the whole quantity is \(Q\), then:
\[
\begin{aligned}
\text{part}
&=
\frac{a}{b}\times Q.
\end{aligned}
\]
The denominator \(b\) cannot be zero.
2. Example: \(3/8\) of 240 kg
The word problem is:
Find \(3/8\) of 240 kg.
Write the multiplication:
\[
\begin{aligned}
\frac{3}{8}\times240
&=
\frac{3\times240}{8}.
\end{aligned}
\]
Multiply and divide:
\[
\begin{aligned}
\frac{3\times240}{8}
&=
\frac{720}{8}\\
&=
90.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{\frac{3}{8}\text{ of }240\text{ kg}=90\text{ kg}}.
\end{aligned}
\]
3. Why “of” means multiply
A fraction tells what part of a whole is selected.
Multiplying the whole by the fraction scales the whole down or up.
\[
\begin{aligned}
\text{selected part}
&=
\text{fraction}\times\text{whole}.
\end{aligned}
\]
For example, \(1/2\) of a quantity is half of it, so multiplying by \(1/2\) gives half the whole.
4. Two common methods
There are two equivalent ways to calculate a fraction of a quantity.
Method A: multiply first
\[
\begin{aligned}
\frac{3}{8}\times240
&=
\frac{3\times240}{8}\\
&=
90.
\end{aligned}
\]
Method B: divide first
\[
\begin{aligned}
\frac{3}{8}\times240
&=
3\times\left(240\div8\right)\\
&=
3\times30\\
&=
90.
\end{aligned}
\]
Dividing first is often easier when the whole quantity is divisible by the denominator.
5. Recipe problems
Recipe problems often ask for a fraction of an ingredient.
For example:
A recipe has 150 g of flour. You need \(2/3\) of it. How much flour do you need?
\[
\begin{aligned}
\frac{2}{3}\times150
&=
100.
\end{aligned}
\]
The answer is \(100\text{ g}\).
6. Discount problems
A discount can be written as a fraction of the original price.
For example, a discount of \(1/4\) on an \$80 item is:
\[
\begin{aligned}
\frac{1}{4}\times80
&=
20.
\end{aligned}
\]
The discount is \$20. The remaining price is:
\[
\begin{aligned}
80-20&=60.
\end{aligned}
\]
So the sale price is \$60.
7. Sharing and group problems
In sharing problems, a fraction tells what part of the total is used, selected, or distributed.
For example:
\[
\begin{aligned}
\frac{5}{6}\times30
&=
25.
\end{aligned}
\]
So \(5/6\) of 30 people is 25 people.
8. Decimal quantities
Decimal quantities can also be used. For example:
\[
\begin{aligned}
\frac{7}{10}\times12.5
&=
8.75.
\end{aligned}
\]
So \(7/10\) of 12.5 L is 8.75 L.
9. Percent connection
Fractions can be converted to percentages.
Since
\[
\begin{aligned}
\frac{3}{8}
&=
0.375
=
37.5\%,
\end{aligned}
\]
finding \(3/8\) of a quantity is the same as finding \(37.5\%\) of that quantity.
10. Checking the answer
A good check is to divide the part by the whole. The result should be the original fraction.
\[
\begin{aligned}
\frac{\text{part}}{\text{whole}}
&=
\frac{90}{240}\\
&=
\frac{3}{8}.
\end{aligned}
\]
Since the check returns \(3/8\), the answer is consistent.
11. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
12. Common mistakes
- Adding the fraction to the quantity instead of multiplying.
- Dividing by the numerator instead of the denominator.
- Forgetting to attach the correct unit to the final answer.
- In discount problems, finding the discount but forgetting to subtract it from the original price.
- Rounding too early when the quantity is a decimal.
- Using a denominator of zero, which is undefined.
Key idea: in fraction word problems, “of” usually means multiply the whole quantity by the fraction.
