Addition and subtraction including decimal numbers
🎯 In this topic you will
- compose, decompose, and regroup numbers to support efficient calculation
- estimate, add, and subtract decimal numbers with accuracy
- use shapes and symbols to represent two unknown numbers in addition and subtraction calculations
🧠 Key Words
- carry
- carrying
- symbol
Show Definitions
- carry: In addition, moving a digit to the next place value when the sum exceeds 9.
- carrying: The process of transferring the extra value to the next column during a column addition calculation.
- symbol: A visual sign (such as a shape or letter) used to represent a number or value in a calculation.
How Gymnastics Scoring Works
In a gymnastics competition, judges give a score for how difficult a routine is and a score for how well it is performed.
Finding the Final Score
The two scores are added together to give the final total.
Working with Decimal Scores
All the scores are decimal numbers.
What This Unit Is About
In this unit, you will add and subtract decimal numbers.
❓ EXERCISES
Exercise $5.1$
1. What decimal number is represented by $70 + 8 + 0.3 + 0.01$?
👀 Show answer
$70 + 8 + 0.3 + 0.01 = 78.31$, so the decimal number is $\boxed{78.31}$.
2. Estimate then calculate.
a. $28.2 + 13.4$
b. $12.46 + 1.31$
c. $13.41 + 4.39$
d. $28.2 - 13.8$
e. $123.1 - 47.3$
Compare your answers with your partner. Did you use the same method?
Check your answers with a calculator.
👀 Show answer
Estimates (to the nearest whole number):
$28.2 \approx 28$, $13.4 \approx 13$ so $28 + 13 \approx 41$.
$12.46 \approx 12$, $1.31 \approx 1$ so $12 + 1 \approx 13$.
$13.41 \approx 13$, $4.39 \approx 4$ so $13 + 4 \approx 17$.
$28.2 \approx 28$, $13.8 \approx 14$ so $28 - 14 \approx 14$.
$123.1 \approx 123$, $47.3 \approx 47$ so $123 - 47 \approx 76$.
Exact answers:
a. $28.2 + 13.4 = 41.6$
b. $12.46 + 1.31 = 13.77$
c. $13.41 + 4.39 = 17.8$
d. $28.2 - 13.8 = 14.4$
e. $123.1 - 47.3 = 75.8$
3. Alana hands in her homework.
$3.4 + 1.8$ and $6.5 - 2.7$ are shown in column layouts.
Mark the homework and correct any errors.
What advice would you give to Alana?
👀 Show answer
Both answers are incorrect.
$3.4 + 1.8 = 5.2$, not $4.12$.
$6.5 - 2.7 = 3.8$, not $4.2$.
Advice: Line up the decimal points carefully so that ones, tenths and hundredths are in the correct columns, then add or subtract each column, using carrying or exchanging where needed.
4. Fatima has $\$7.25$. She is given $\$15.50$. How much does she have now?
👀 Show answer
$\$7.25 + \$15.50 = \$22.75$, so Fatima now has $\boxed{\$22.75}$.
5. In a sale, a shop takes $\$2.25$ off the price of these books.
a. What is the cost of each book in the sale?
b. What is the total cost of the four books in the sale?
👀 Show answer
The marked prices of the books are $\$6.65$, $\$16.35$, $\$15.50$ and $\$8.70$.
Sale price for each book is original price minus $\$2.25$:
Book A: $\$6.65 - \$2.25 = \$4.40$
Book B: $\$16.35 - \$2.25 = \$14.10$
Book C: $\$15.50 - \$2.25 = \$13.25$
Book D: $\$8.70 - \$2.25 = \$6.45$
Total sale cost: $\$4.40 + \$14.10 + \$13.25 + \$6.45 = \$38.20$.
6.
a. The square and the circle each stand for a different whole number.
Square $+ $ square $+ $ square $= 42$
Square $+ $ circle $= 23$
What is the value of each shape? Compare your answer with your partner.
b. The triangle and the circle each stand for a different whole number.
Triangle $+ $ triangle $= 18$
Triangle $- $ circle $= 5$
What is the value of each shape?
👀 Show answer
6a. Let the square be $s$ and the circle be $c$.
From $s + s + s = 42$ we get $3s = 42$, so $s = 14$.
From $s + c = 23$ we get $14 + c = 23$, so $c = 9$.
So the square is $14$ and the circle is $9$.
6b. Let the triangle be $t$ and the circle be $c$.
From $t + t = 18$ we get $2t = 18$, so $t = 9$.
From $t - c = 5$ we get $9 - c = 5$, so $c = 4$.
So the triangle is $9$ and the circle is $4$.
7. Each symbol stands for a number. Find the value of each symbol.
👀 Show answer
The first column has three squares and a total of $45$, so $3s = 45$ and $s = 15$.
The top row has a square and two circles with total $27$, so $15 + 2c = 27$ giving $2c = 12$ and $c = 6$.
The bottom row has a square, a circle and a triangle with total $28$, so $15 + 6 + t = 28$ giving $t = 7$.
So the square is $15$, the circle is $6$ and the triangle is $7$.
8. What are the possible values of the square and the triangle when square $+$ square $+$ square $+$ triangle $= 1.3\,\text{kg}$?
👀 Show answer
Not enough information provided to determine a unique value.
🧠 Think like a Mathematician
A magic square
Place the numbers $0.1$, $0.2$, $0.3$, $0.4$, $0.5$, $0.6$, $0.7$, $0.8$ and $0.9$ so that the total of each row, column and diagonal is $1.5$.
You will show you are specialising when you find a solution to the problem.
Follow-up Questions:
👀 show answer
One possible magic square is:
$0.8 \quad 0.1 \quad 0.6$
$0.3 \quad 0.5 \quad 0.7$
$0.4 \quad 0.9 \quad 0.2$
Each row, column and diagonal in this arrangement adds to $1.5$.
👀 show answer
In a $3 \times 3$ magic square, the middle number is the average of all the numbers used.
The numbers from $0.1$ to $0.9$ increase by $0.1$ each time, so the middle value is the mean: $\dfrac{0.1 + 0.9}{2} = 0.5$. This must sit in the centre so that every row, column and diagonal can balance to the same total.
👀 show answer
All other solutions are rotations or reflections of one basic magic square.
If you rotate the square by $90^\circ$, $180^\circ$, or $270^\circ$, or reflect it across a mirror line, you get a new arrangement that still uses the same numbers and keeps every row, column and diagonal adding to $1.5$.