Calculating with fractions
🎯 In this topic you will
- Estimate and add fractions with the same denominator (within one whole).
- Estimate and subtract fractions with the same denominator (within one whole).
- Solve a range of fraction problems using these skills.
🧠 Key Words
- decreased
- original
Show Definitions
- decreased: Became smaller in value or amount; in mathematics, this means a quantity has been reduced.
- original: The starting value or amount before any changes such as adding, subtracting, or decreasing.
🔢 Fractions Are Numbers Too
You can add and subtract fractions because fractions are numbers too. Finding a fraction of a whole has the effect of decreasing the original value.
🛍️ How Sales Reduce Prices
In a sale, prices are decreased so that you can buy what you wish for less.
❓ EXERCISES
1. What addition is shown on this diagram?
👀 Show answer
2. Use diagrams or fraction strips to help you complete each addition. Estimate before you calculate. Draw a ring around your estimate.
a. $\dfrac{1}{3}+\dfrac{1}{3}=$ estimate: $<\dfrac{1}{2},=\dfrac{1}{2},>\dfrac{1}{2}$
b. $\dfrac{2}{5}+\dfrac{2}{5}=$ estimate: $<\dfrac{1}{2},=\dfrac{1}{2},>\dfrac{1}{2}$
c. $\dfrac{2}{10}+\dfrac{3}{10}=$ estimate: $<\dfrac{1}{2},=\dfrac{1}{2},>\dfrac{1}{2}$
d. $\dfrac{1}{2}+\dfrac{1}{4}=$ estimate: $<\dfrac{1}{2},=\dfrac{1}{2},>\dfrac{1}{2}$
👀 Show answer
b. $\dfrac{4}{5}$ (estimate $>\dfrac{1}{2}$)
c. $\dfrac{5}{10}=\dfrac{1}{2}$ (estimate $=\dfrac{1}{2}$)
d. $\dfrac{3}{4}$ (estimate $>\dfrac{1}{2}$)
3. Find all the possible solutions for this calculation.
$\dfrac{\square}{4}+\dfrac{\square}{4}=1$
👀 Show answer
4. What subtraction is shown on this diagram?
👀 Show answer
5. Use diagrams or fraction strips to help you complete each subtraction. Estimate before you calculate. Draw a ring around your estimate.
a. $\dfrac{2}{3}-\dfrac{2}{3}=$ estimate: $<\dfrac{1}{2},=\dfrac{1}{2},>\dfrac{1}{2}$
b. $\dfrac{3}{5}-\dfrac{2}{5}=$ estimate: $<\dfrac{1}{2},=\dfrac{1}{2},>\dfrac{1}{2}$
c. $1-\dfrac{3}{10}=$ estimate: $<\dfrac{1}{2},=\dfrac{1}{2},>\dfrac{1}{2}$
d. $1-\dfrac{3}{4}=$ estimate: $<\dfrac{1}{2},=\dfrac{1}{2},>\dfrac{1}{2}$
👀 Show answer
b. $\dfrac{1}{5}$ (estimate $<\dfrac{1}{2}$)
c. $\dfrac{7}{10}$ (estimate $>\dfrac{1}{2}$)
d. $\dfrac{1}{4}$ (estimate $<\dfrac{1}{2}$)
6. Find all the possible solutions for this calculation.
$1-\dfrac{\square}{5}=\dfrac{\square}{5}$
👀 Show answer
🧠 Think like a Mathematician
Hien explored adding thirds to make $1$ and subtracting thirds from $1$. He made this table.
| Thirds | |
|---|---|
| Addition to make $1$ | Subtraction from $1$ |
| $\dfrac{0}{3}+\dfrac{3}{3}=\dfrac{3}{3}=1$ | $1-\dfrac{3}{3}=\dfrac{3}{3}=0$ |
| $\dfrac{1}{3}+\dfrac{2}{3}=\dfrac{3}{3}=1$ | $1-\dfrac{2}{3}=\dfrac{1}{3}$ |
| $1-\dfrac{1}{3}=\dfrac{2}{3}$ | |
| $1-\dfrac{0}{3}=\dfrac{3}{3}=1$ | |
Hien says, “There are twice as many subtractions as there are additions. I wonder if that is true for other fractions too?” What do you think. Give some examples.
Show Answers
- Pattern: For thirds, there are $2$ ways to add to make $1$ but $4$ subtraction results from $1$ (using $0/3,1/3,2/3,3/3$).
- Why: When adding to make $1$, only pairs that sum to the denominator work (for thirds: $0+3$ and $1+2$). For subtraction, every possible numerator from $0$ to the denominator gives a result.
- Example (fifths): Additions to make $1$: $0/5+5/5$, $1/5+4/5$, $2/5+3/5$ → $3$ additions. Subtractions from $1$: $1-n/5$ for $n=0,1,2,3,4,5$ → $6$ subtractions.
- Conclusion: Yes — for fractions with denominator $d$, there are $\dfrac{d+1}{2}$ addition pairs (rounded down) but $d+1$ subtraction results. So subtraction cases are always about twice as many.
❓ EXERCISES
7. A clothes shop is having a sale. All prices are reduced by $\dfrac{1}{4}$. How much does each item cost now?
👀 Show answer
a. $\dfrac{3}{4}\times 8=6$
b. $\dfrac{3}{4}\times 4=3$
c. $\dfrac{3}{4}\times 32=24$
d. $\dfrac{3}{4}\times 20=15$