Fractions are used whenever a whole quantity is split, scaled, compared, or shared.
Common examples include recipes, ratios, proportions, discounts, measurements, grades, and group sharing.
1. Recipe scaling
If a recipe is written for one number of servings and you need a different number,
multiply each ingredient by a scale factor:
\[
\begin{aligned}
\text{scale factor}
&=
\frac{\text{target servings}}{\text{original servings}}.
\end{aligned}
\]
Then:
\[
\begin{aligned}
\text{new ingredient amount}
&=
\text{old ingredient amount}\times\text{scale factor}.
\end{aligned}
\]
Example: a recipe for 6 people uses \(2\frac{1}{2}\) cups. Scale it for 9 people.
\[
\begin{aligned}
\text{scale factor}
&=
\frac{9}{6}
=
\frac{3}{2},\\
2\frac{1}{2}\times\frac{3}{2}
&=
\frac{5}{2}\times\frac{3}{2}\\
&=
\frac{15}{4}\\
&=
3\frac{3}{4}.
\end{aligned}
\]
The scaled ingredient amount is \(3\frac{3}{4}\) cups.
2. Ratio sharing
A ratio such as \(2:3\) means the total is split into \(2+3=5\) equal ratio parts.
\[
\begin{aligned}
\text{first share}
&=
\text{total}\times\frac{2}{2+3},\\
\text{second share}
&=
\text{total}\times\frac{3}{2+3}.
\end{aligned}
\]
Example: share 150 g in the ratio \(2:3\).
\[
\begin{aligned}
\text{first share}
&=
150\times\frac{2}{5}
=
60,\\
\text{second share}
&=
150\times\frac{3}{5}
=
90.
\end{aligned}
\]
3. Equal sharing
Equal sharing means dividing the total quantity by the number of people or groups.
\[
\begin{aligned}
\text{one share}
&=
\frac{\text{total quantity}}{\text{number of shares}}.
\end{aligned}
\]
Example: share \(7\frac{1}{2}\) kg among 5 people.
\[
\begin{aligned}
7\frac{1}{2}\div5
&=
\frac{15}{2}\div5\\
&=
\frac{15}{2}\times\frac{1}{5}\\
&=
\frac{3}{2}\\
&=
1\frac{1}{2}.
\end{aligned}
\]
Each person gets \(1\frac{1}{2}\) kg.
4. Proportions
A proportion states that two ratios are equal:
\[
\begin{aligned}
\frac{a}{b}
&=
\frac{c}{x}.
\end{aligned}
\]
Cross multiplication gives:
\[
\begin{aligned}
ax&=bc.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
x&=\frac{bc}{a},
\qquad a\ne0.
\end{aligned}
\]
Example:
\[
\begin{aligned}
\frac{3}{4}
&=
\frac{9}{x}.
\end{aligned}
\]
Then:
\[
\begin{aligned}
3x&=4\cdot9,\\
x&=12.
\end{aligned}
\]
5. Fraction of a quantity
Many word problems use the word of. In fraction problems, “of” usually means multiplication.
\[
\begin{aligned}
\text{part}
&=
\text{fraction}\times\text{whole}.
\end{aligned}
\]
Example:
\[
\begin{aligned}
\frac{3}{8}\text{ of }240
&=
\frac{3}{8}\times240\\
&=
90.
\end{aligned}
\]
6. Choosing the correct template
A good first step in a real-life fraction problem is identifying the situation:
- Recipe scaling: the number of servings changes.
- Ratio sharing: a total is split according to ratio parts.
- Equal sharing: a total is divided equally.
- Proportion: two ratios are equal and one value is missing.
- Fraction of a quantity: the problem asks for part of a whole.
7. Exact fraction arithmetic
It is often better to keep fractions exact during the calculation.
Decimal approximations are useful for practical measurement, but exact fractions prevent rounding errors.
\[
\begin{aligned}
\frac{5}{2}\times\frac{3}{2}
&=
\frac{15}{4}
=
3\frac{3}{4}.
\end{aligned}
\]
8. Checking answers
Different scenarios have different checks:
- Recipe: scaled amount divided by old amount should equal the scale factor.
- Ratio: shares should add back to the total.
- Sharing: one share multiplied by the number of shares should return the total.
- Proportion: cross products should match.
- Fraction of a quantity: part divided by whole should return the original fraction.
9. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
10. Common mistakes
- Using the scale factor upside down in recipe problems.
- Forgetting to add ratio parts before finding each share.
- Dividing by the numerator instead of the denominator.
- Rounding too early before the final answer.
- Forgetting units in the final answer.
- Setting up a proportion incorrectly by mismatching units or categories.
Key idea: identify the real-life situation first, then choose the matching fraction model.
