Multiplying a decimal and a whole number
🎯 In this topic you will
- Multiply a number with 1 decimal place by a whole number.
🧠 Key Words
- decimal
- decimal place
- decimal point
Show Definitions
- decimal: A number that includes a fractional part separated from the whole number using a decimal point.
- decimal place: The position of a digit to the right of the decimal point, showing the value of that part of the number.
- decimal point: The dot in a number that separates the whole number part from the fractional part.
Looking at the Picture
L ook at the picture.
The Girl’s Height
T he girl is 90 cm tall.
Comparing Heights
T he girl is twice as tall as the cat.
Understanding Halves
T he cat is half or 0.5 times the height of the girl.
Finding Half of 90 cm
H alf of 90 cm is 45 cm.
Calculating with 0.5
Z ero point five times 90 equals 45 cm.
Height of the Cat
T he cat is 45 cm tall.
What You Will Learn Next
I n this section you will learn to multiply numbers with one decimal place by whole numbers.
❓ EXERCISES
1. Sofia counts in steps of zero-point-three. She says, 'zero-point-three, zero-point-six, ...'. She continues counting to find $4 \times 0.3$. What is her answer?
👀 Show answer
Each step adds $0.3$. Counting the steps gives the sequence $0.3, 0.6, 0.9, 1.2$. The fourth term is $1.2$, so $4 \times 0.3 = 1.2$.
2. Draw a number line to help you calculate.
a. $7 \times 0.4$
b. $5 \times 0.5$
c. $3 \times 0.9$
👀 Show answer
On a number line, each jump represents one multiple of the decimal.
For $7 \times 0.4$: seven jumps of $0.4$ give $7 \times 0.4 = 2.8$.
For $5 \times 0.5$: five jumps of $0.5$ give $5 \times 0.5 = 2.5$.
For $3 \times 0.9$: three jumps of $0.9$ give $3 \times 0.9 = 2.7$.
3. A, B and C stand for missing numbers on this multiplication grid. Find the value of A, B and C.
| $ \times $ | $6$ | $5$ | $A$ |
|---|---|---|---|
| $0.5$ | $B$ | $2.5$ | $2$ |
| $0.2$ | $1.2$ | $C$ | $0.8$ |
| $0.6$ | $3.6$ | $3$ | $2.4$ |
👀 Show answer
Each entry is the product of its row and column headings.
From the last row: $0.6 \times A = 2.4$, so $A = \dfrac{2.4}{0.6} = 4$.
From the first row, second column: $0.5 \times 6 = B$, so $B = 3$.
From the second row, third column: $0.2 \times 5 = C$, so $C = 1$.
Therefore, $A = 4$, $B = 3$ and $C = 1$.
4. Copy and complete these calculations.
a. $0.8 \times 9$
b. $1.3 \times 7$
👀 Show answer
For part (a): $0.8 = \dfrac{8}{10}$, so
$0.8 \times 9 = \dfrac{8}{10} \times 9 = \dfrac{8 \times 9}{10} = \dfrac{72}{10} = 7.2$.
For part (b): $1.3 = \dfrac{13}{10}$, so
$1.3 \times 7 = \dfrac{13}{10} \times 7 = \dfrac{13 \times 7}{10} = \dfrac{91}{10} = 9.1$.
Therefore, $0.8 \times 9 = 7.2$ and $1.3 \times 7 = 9.1$.
5. Calculate.
a. $0.6 \times 7$
b. $1.4 \times 5$
👀 Show answer
For $0.6 \times 7$, think of $0.6$ as $\dfrac{6}{10}$:
$0.6 \times 7 = \dfrac{6}{10} \times 7 = \dfrac{42}{10} = 4.2$.
For $1.4 \times 5$, think of $1.4$ as $\dfrac{14}{10}$:
$1.4 \times 5 = \dfrac{14}{10} \times 5 = \dfrac{70}{10} = 7$.
So the answers are $4.2$ and $7$.
🧠 Reasoning Tip
Estimate your answer before you calculate it. You can use diagrams similar to those in question $4$ to help you.
6. Which calculation is the odd one out? Explain why.
$1.4 \times 5 \quad\quad 3.5 \times 7 \quad\quad 2.5 \times 8 \quad\quad 1.8 \times 5 \quad\quad 3.5 \times 6$
Check your answers with your partner. Did you remember to estimate before you calculated the answer?
👀 Show answer
Calculate each product:
$1.4 \times 5 = 7$
$3.5 \times 7 = 24.5$
$2.5 \times 8 = 20$
$1.8 \times 5 = 9$
$3.5 \times 6 = 21$
All of the answers except $3.5 \times 7$ are whole numbers. $3.5 \times 7 = 24.5$, which is not a whole number, so $3.5 \times 7$ is the odd one out.
7. What is the product of $15.4$ and $7$?
👀 Show answer
Write $15.4$ as $\dfrac{154}{10}$:
$15.4 \times 7 = \dfrac{154}{10} \times 7 = \dfrac{154 \times 7}{10} = \dfrac{1078}{10} = 107.8$.
The product of $15.4$ and $7$ is $107.8$.
8. Copy and complete these calculations.
a. Grid for multiplying a decimal by $7$.
b. Grid for multiplying a decimal by $4$.
c. Grid for multiplying a decimal by $5$.
👀 Show answer
The first grid represents $1.6 \times 7$:
$1.6 \times 7 = \dfrac{16}{10} \times 7 = \dfrac{112}{10} = 11.2$.
The second grid represents $7.3 \times 4$:
$7.3 \times 4 = \dfrac{73}{10} \times 4 = \dfrac{292}{10} = 29.2$.
The third grid represents $26.2 \times 5$:
$26.2 \times 5 = \dfrac{262}{10} \times 5 = \dfrac{1310}{10} = 131$.
So the completed products are $11.2$, $29.2$ and $131$ respectively.
🧠 Think like a Mathematician
You need these cards:
3546
Arrange the cards as a multiplication calculation.
Investigate different answers. Which one is the biggest? Which one is the smallest? How many different answers can you find?
You will show you are specialising when you find solutions to the problem.
Your Task:
- Create every possible two-digit × one-digit multiplication you can using the cards 3, 5, 4, 6.
- Calculate each product.
- List all distinct answers.
- Identify the largest and the smallest product.
- Explain why those products are largest/smallest based on place value.
Follow-up Questions:
Show Answers
- 1: The biggest product occurs when the largest digit is placed in the tens place. Using $6$ in the tens place and multiplying by the next largest card: $64 \times 5 = 320$ is the biggest possible answer.
- 2: The smallest product occurs when the smallest digit is in the tens place and you multiply by the next smallest value: $34 \times 5 = 170$ (or $35 \times 4 = 140$ depending on arrangement). The smallest of all is $34 \times 4 = 136$.
- 3: There are many possible combinations. With four cards where two form a two-digit number and the other is the multiplier, there are $12$ possible arranged products, though some may repeat. Distinct answers: $136, 140, 150, 160, 168, 180, 192, 200, 210, 240, 252, 320$.
- 4: Products are largest when the biggest digit is placed in the tens place. This is because increasing the tens digit increases the value of the entire multiplicand by a factor of ten. Place value, not just the digits themselves, determines the scale of the product.