Multiplying an integer by a mixed number
🎯 In this topic you will
- Learn how to multiply an integer by a mixed number
🧠 Key Words
- mean
- partitioning
- simplified
Show Definitions
- mean: The average of a set of numbers, calculated by adding all values and dividing by the count of values.
- partitioning: The process of dividing something into separate parts or sections, often used in mathematics and computer science to break down complex problems.
- simplified: Made less complex or complicated by removing unnecessary elements while preserving essential characteristics or functions.
❓ EXERCISES
1. Complete the following multiplications:
a. $3\frac{1}{2} \times 8$
b. $2\frac{2}{3} \times 6$
c. $1\frac{3}{4} \times 4$
d. $5\frac{1}{5} \times 10$
👀 Show answer
a. $3\frac{1}{2} \times 8 = 3 \times 8 + \frac{1}{2} \times 8 = 24 + 4 = 28$
b. $2\frac{2}{3} \times 6 = 2 \times 6 + \frac{2}{3} \times 6 = 12 + 4 = 16$
c. $1\frac{3}{4} \times 4 = 1 \times 4 + \frac{3}{4} \times 4 = 4 + 3 = 7$
d. $5\frac{1}{5} \times 10 = 5 \times 10 + \frac{1}{5} \times 10 = 50 + 2 = 52$
2. A rectangle is $15$ m long and $2\frac{1}{3}$ m wide. Calculate the area of the rectangle.
a. Estimate the area.
b. Calculate the exact area.
👀 Show answer
a. Estimate: $15 \times 2 = 30$ m²
b. Exact: $15 \times 2\frac{1}{3} = 15 \times \frac{7}{3} = \frac{15 \times 7}{3} = \frac{105}{3} = 35$ m²
3. Lin has $20$ containers with an average capacity of $2\frac{2}{5}$ liters. Use the formula "Total = Average × Number of containers" to determine if the total capacity is $46$ liters. Explain your reasoning.
👀 Show answer
Total capacity = $2\frac{2}{5} \times 20 = \frac{12}{5} \times 20 = \frac{12 \times 20}{5} = \frac{240}{5} = 48$ liters
The total capacity is $48$ liters, not $46$ liters.
4. Complete the following multiplications by breaking each mixed number into its whole number and fractional parts:
💡 Quick Math Tip
Estimation with mixed numbers: When estimating, round mixed numbers like 4½ to the nearest whole number (5) before multiplying.
a. $3\frac{1}{4} \times 8$
b. $2\frac{3}{5} \times 10$
c. $4\frac{1}{2} \times 6$
d. $1\frac{7}{8} \times 4$
👀 Show answer
a. $3\frac{1}{4} \times 8 = 3 \times 8 + \frac{1}{4} \times 8 = 24 + 2 = 26$
b. $2\frac{3}{5} \times 10 = 2 \times 10 + \frac{3}{5} \times 10 = 20 + 6 = 26$
c. $4\frac{1}{2} \times 6 = 4 \times 6 + \frac{1}{2} \times 6 = 24 + 3 = 27$
d. $1\frac{7}{8} \times 4 = 1 \times 4 + \frac{7}{8} \times 4 = 4 + 3\frac{1}{2} = 7\frac{1}{2}$
5. The diagram shows a square joined to a rectangle. Work out the area of the shape.
a. An estimate for the area of the shape.
b. The accurate area of the shape.
👀 Show answer
a. Estimate: Square area ≈ $5 \times 5 = 25$ cm², Rectangle area ≈ $5 \times 13 = 65$ cm², Total area ≈ $25 + 65 = 90$ cm²
b. Accurate: Square area = $5 \times 5 = 25$ cm², Rectangle area = $5 \times 12\frac{4}{9} = 5 \times \frac{112}{9} = \frac{560}{9} = 62\frac{2}{9}$ cm², Total area = $25 + 62\frac{2}{9} = 87\frac{2}{9}$ cm²
6. Martha is paving a rectangular area in her garden. The rectangle is $3\frac{3}{4}$ meters long and $2$ meters wide.
a. Is the estimated area of the garden correct? Explain your reasoning.
b. Calculate the actual area of the rectangular garden.
c. The paving tiles cost $42$ yuan per square meter. Martha needs to buy whole square meters of tiles, and she says the total cost is $294$ yuan. Is this statement correct? Explain your reasoning.
👀 Show answer
a. The estimated area of 8 square meters is not correct. The actual area is 7.5 square meters.
b. Rectangular area = length × width = $3\frac{3}{4} \times 2 = \frac{15}{4} \times 2 = \frac{30}{4} = 7.5$ square meters.
c. Martha's statement is not correct. The actual area is 7.5 square meters, so she needs to buy 8 square meters (whole square meters). Total cost = $8 \times 42 = 336$ yuan, not 294 yuan.
🧠 Think like a Mathematician
Question: Compare Anders' and Xavier's methods for calculating $3\frac{2}{3} \times 8$. What are the advantages and disadvantages of each approach?
Equipment: Paper, pencil, calculator
Method:
- Examine Anders' method: Breaking down the mixed fraction into $3 \times 8 + \frac{2}{3} \times 8$
- Examine Xavier's method: Converting to improper fraction $\frac{11}{3}$ and then multiplying by 8
- Analyze the advantages and disadvantages of each method
- Apply one of the methods to solve a new problem
- Reflect on which method you prefer and why
Follow-up Questions:
👀 Show Answers
- a) Anders' method: Advantage - Works with whole numbers and fractions separately, which can be more intuitive. Disadvantage - May involve more steps. Xavier's method: Advantage - Streamlined approach with fewer steps. Disadvantage - Requires comfort with converting mixed numbers to improper fractions.
- b) Using Anders' method: $4 \times 6 + \frac{1}{2} \times 6 = 24 + 3 = 27$. Using Xavier's method: $\frac{9}{2} \times 6 = \frac{54}{2} = 27$.
- c) Preference depends on individual comfort with fractions. Some may prefer Anders' method for its step-by-step approach, while others may prefer Xavier's method for its efficiency once comfortable with improper fractions.
❓ EXERCISES
8. This is how Zara works out $2\dfrac{1}{6}\times 15$.
Sofia says, ‘You changed $\dfrac{15}{6}$ to a mixed number and then simplified $\dfrac{3}{6}$ to $\dfrac{1}{2}$. I would have simplified $\dfrac{15}{6}$ to $\dfrac{5}{2}$ before changing it to a mixed number.’
a. Do you prefer Zara’s method or Sofia’s method? Explain why.
b. Use your preferred method to work these out. Write your answer in its simplest form.
i. $3\dfrac{3}{8}\times 10$
ii. $4\dfrac{3}{4}\times 14$
iii. $2\dfrac{7}{10}\times 12$
👀 Show answer
a. Sofia’s method is usually more efficient, because simplifying early avoids unnecessary steps. Zara’s method also works, but it takes longer.
b.
i. $3\dfrac{3}{8}\times 10 = \dfrac{27}{8}\times 10 = \dfrac{270}{8} = \dfrac{135}{4} = 33\dfrac{3}{4}$
ii. $4\dfrac{3}{4}\times 14 = \dfrac{19}{4}\times 14 = \dfrac{266}{4} = \dfrac{133}{2} = 66\dfrac{1}{2}$
iii. $2\dfrac{7}{10}\times 12 = \dfrac{27}{10}\times 12 = \dfrac{324}{10} = \dfrac{162}{5} = 32\dfrac{2}{5}$
9. Jamal works in a garden centre.
It takes him $5\dfrac{1}{4}$ minutes to plant one tray of seedlings.
How long will it take him to plant $50$ trays of seedlings?
Give your answer in hours and minutes.
🔎 Reasoning Tip
Seedlings: Seedlings are seeds that are just starting to grow into plants.
👀 Show answer
Total time $= 5\dfrac{1}{4}\times 50 = \dfrac{21}{4}\times 50 = \dfrac{1050}{4} = 262\dfrac{1}{2}$ minutes.
$262\dfrac{1}{2}$ minutes $= 4$ hours $22\dfrac{1}{2}$ minutes $\approx 4$ hours $23$ minutes.
🧠 Think like a Mathematician
Task: Work through the following questions to explore multiplication with mixed numbers and discover patterns.
Questions:
i. $3\dfrac{1}{5} \times 2$
ii. $3\dfrac{1}{5} \times 3$
👀 show answer
- a i.$3\dfrac{1}{5} = \dfrac{16}{5}$, so $\dfrac{16}{5}\times 2 = \dfrac{32}{5} = 6\dfrac{2}{5}$.
- a ii.$\dfrac{16}{5}\times 3 = \dfrac{48}{5} = 9\dfrac{3}{5}$.
- b. Multiply by 5 (since denominator is 5). $3\dfrac{1}{5}\times 5 = 16$.
- c. - For $3\dfrac{2}{5} = \dfrac{17}{5}$, multiply by 5 → 17. - For $3\dfrac{3}{5} = \dfrac{18}{5}$, multiply by 5 → 18. - For $3\dfrac{4}{5} = \dfrac{19}{5}$, multiply by 5 → 19.
- d. In each case, multiplying by the denominator (5) gives a whole number.
- e. - $3\dfrac{1}{7}=\dfrac{22}{7}$, multiply by 7 → 22. - $3\dfrac{2}{7}=\dfrac{23}{7}$, multiply by 7 → 23. - $3\dfrac{3}{7}=\dfrac{24}{7}$, multiply by 7 → 24. - $3\dfrac{4}{7}=\dfrac{25}{7}$, multiply by 7 → 25. - $3\dfrac{5}{7}=\dfrac{26}{7}$, multiply by 7 → 26. - $3\dfrac{6}{7}=\dfrac{27}{7}$, multiply by 7 → 27.
Pattern: multiplying by the denominator (7) clears the fraction. - f. For other denominators, the same rule applies: multiplying by the denominator of the fractional part always produces a whole number. For example, $2\dfrac{1}{6}=\dfrac{13}{6}$, multiply by 6 → 13. The pattern holds universally.
❓ EXERCISES
11. Work out
a. an estimate for the area of the blue section of this rectangle
b. the accurate area of the blue section of this rectangle.
👀 Show answer
a. Round lengths to the nearest metre: $12\dfrac{3}{5}\!\approx\!13$, $4\dfrac{2}{3}\!\approx\!5$. Blue length $\approx 13-5=8\ \text{m}$. Height $=2\ \text{m}$. Estimated area $\approx 2\times 8=16\ \text{m}^2$.
b. Blue length $=12\dfrac{3}{5}-4\dfrac{2}{3}=\dfrac{63}{5}-\dfrac{14}{3}=\dfrac{189-70}{15}=\dfrac{119}{15}=7\dfrac{14}{15}\ \text{m}$. Accurate area $=2\times\dfrac{119}{15}=\dfrac{238}{15}=15\dfrac{13}{15}\ \text{m}^2$.
⚠️ Be careful!
- Estimate vs exact: Round only for the estimate. Do not use the rounded number in the exact calculation.
Example: estimate $2\dfrac{1}{2}\times16 \approx 3\times16=48$, but exact is $(2\times16)+\left(\tfrac{1}{2}\times16\right)=32+8=40$. - Multiply the whole and fractional parts (or convert first). Never multiply just the whole part.
$4\dfrac{1}{2}\times6=(4\times6)+\left(\tfrac{1}{2}\times6\right)=24+3=27$. - Improper‑fraction method with cancellation is often quicker.
$2\dfrac{2}{5}\times20=\tfrac{12}{5}\times20=12\times\tfrac{20}{5}=12\times4=48$. - Area problems: Use $\text{Area}=\text{length}\times\text{width}$. Keep exact dimensions for the exact area and include square units (e.g., m$^2$).
- “Average × number = total” (don’t forget to apply it to the entire mixed number).
$2\dfrac{2}{5}\times20=\tfrac{12}{5}\times20=48$ (so a claim of $46$ L would be incorrect). - Simplify the final answer and write as a mixed number when appropriate.