Primary & Middle School Mathematics · Grades 1–9

30Computing with Decimals

Decimal numbers add, subtract and multiply almost like whole numbers — the whole art is knowing where the point goes. This chapter covers the column methods and the magic of multiplying or dividing by $10$ , $100$ , $1000$ . (Multiplying two decimals together waits for Chapter 38.)

30.1 Adding and subtracting

Method 30.1 (Columns with a point)

Write the numbers with the decimal points aligned (then units are under units, tenths under tenths);
pad the shorter decimal part with zeros;
add or subtract as with whole numbers;
drop the point of the result straight down.

Example 30.2

$27.5 + 3.86$ (pad $27.5 = 27.50$ ):

\begin{array}{r} 2\,7.5\,0 \\ +\ \ \ 3.8\,6 \\ \hline 3\,1.3\,6 \end{array} \qquad\qquad \begin{array}{r} 1\,2.4\,0 \\ -\ \ \ 5.6\,5 \\ \hline \ \ \,6.7\,5 \end{array}

For the subtraction ( $12.4 - 5.65$ ): pad, borrow as usual, and check by adding back: $6.75 + 5.65 = 12.4$ .

Example 30.3 (Mental complements)

What must be added to $7.6$ to reach $10$ ? Climb: $7.6 \to 8$ is $0.4$ , then $8 \to 10$ is $2$ : the complement is $2.4$ . Handy for change: pay $10$ for a $7.60$ item, get $2.40$ back.

30.2 Ten times bigger, ten times smaller

Proposition 30.4 (Times and divided by 10, 100, 1000)

Multiplying by $10$ makes each digit worth ten times more: the digits slide one place left — which looks like the point moving one place right. Dividing does the opposite:

3.47 \times 10 = 34.7, \qquad 3.47 \times 100 = 347, \qquad 52.1 \div 10 = 5.21, \qquad 6 \div 100 = 0.06 .

Proof. Admitted at this level. ∎

Multiplying by 10: every digit climbs one place to the left. The point does not really move — the digits do. — Multiplying by $10$ : every digit climbs one place to the left. The point does not really move — the digits do.

Example 30.5 (Conversions again)

Units of measure are powers of ten apart, so conversions are just these shifts: $2.35$ m $= 235$ cm ( $\times 100$ ); $470$ g $= 0.47$ kg ( $\div 1000$ ); $1.5$ L $= 150$ cL.

30.3 Multiplying a decimal by a whole number

Method 30.6 (Decimal times whole)

Compute as if there were no point, then give the result as many decimal digits as the decimal factor:

4.35 \times 6: \quad 435 \times 6 = 2\,610 \ \to\ 4.35 \times 6 = 26.10 = 26.1 .

Estimate to place the point with confidence: $4.35$ is close to $4$ , and $4 \times 6 = 24$ — so $26.1$ , not $2.61$ nor $261$ .

Example 30.7 (Repeated addition still works)

$0.75 \times 4 = 0.75 + 0.75 + 0.75 + 0.75 = 3$ : four quarters make a whole, in decimal clothing. Multiplying by a whole number is still “so many times”.

Example 30.8 (Shopping)

Three notebooks at $2.45$ each and a pen at $1.20$ :

notebooks: $2.45 \times 3 = 7.35$ ;
total: $7.35 + 1.20 = 8.55$ ;
change from $10$ : $10 - 8.55 = 1.45$ .

30.4 Exercises

Exercise 30.1 ★

Compute in columns: $45.7 + 8.93$ ; $16.25 + 3.75$ ; $0.86 + 12.4$ .

Solution

Solution of Exercise 30.1.

$45.7 + 8.93 = 54.63$ ; $16.25 + 3.75 = 20$ ; $0.86 + 12.4 = 13.26$ .

Exercise 30.2 ★

Compute in columns and check: $9.4 - 2.72$ ; $20 - 13.45$ ; $6.03 - 5.9$ .

Solution

Solution of Exercise 30.2.

$9.4 - 2.72 = 6.68$ (check: $6.68 + 2.72 = 9.4$ ).

$20 - 13.45 = 6.55$ (check: $6.55 + 13.45 = 20$ ).

$6.03 - 5.9 = 0.13$ .

Exercise 30.3 ★

Mental complements to $10$ (as in Example 30.3): $6.2$ ; $3.75$ ; $9.99$ .

Solution

Solution of Exercise 30.3.

$6.2 \to 10$ : $3.8$ . $3.75 \to 10$ : $6.25$ . $9.99 \to 10$ : $0.01$ .

Exercise 30.4 ★

Compute without columns:

7.24 \times 10, \qquad 0.58 \times 100, \qquad 34.5 \div 10, \qquad 7 \div 100, \qquad 0.3 \times 1000 .

Solution

Solution of Exercise 30.4.

$72.4$ ; $58$ ; $3.45$ ; $0.07$ ; $300$ .

Exercise 30.5 ★

Convert: $4.05$ m in cm; $325$ cm in m; $0.6$ kg in g; $85$ cL in L.

Solution

Solution of Exercise 30.5.

$4.05$ m $= 405$ cm; $325$ cm $= 3.25$ m; $0.6$ kg $= 600$ g; $85$ cL $= 0.85$ L.

Exercise 30.6 ★

Estimate first, then compute: $6.2 \times 4$ ; $3.45 \times 8$ ; $12.5 \times 6$ .

Solution

Solution of Exercise 30.6.

$6.2 \times 4$ : estimate $24$ ; exactly $24.8$ .

$3.45 \times 8$ : estimate $28$ ( $3.5 \times 8$ ); exactly $27.6$ .

$12.5 \times 6$ : estimate $75$ ; exactly $75$ .

Exercise 30.7 ★

One lap of a track measures $0.4$ km. How long are $7$ laps? And $25$ laps?

Solution

Solution of Exercise 30.7.

$7$ laps: $0.4 \times 7 = 2.8$ km. $25$ laps: $0.4 \times 25 = 10$ km.

Exercise 30.8 ★

Lea buys $2$ baguettes at $1.15$ each and a cake at $12.60$ . She pays with a $20$ bill. Compute her total and her change.

Solution

Solution of Exercise 30.8.

Baguettes: $1.15 \times 2 = 2.30$ . Total: $2.30 + 12.60 = 14.90$ . Change: $20 - 14.90 = 5.10$ .

Exercise 30.9 ★

A bottle holds $1.5$ L. How many liters in a pack of $6$ bottles? A family drinks $0.75$ L per day: for how many days does one pack last? (How many times does $0.75$ fit in $9$ ?)

Solution

Solution of Exercise 30.9.

Pack: $1.5 \times 6 = 9$ L. Days: $0.75 + 0.75 = 1.5$ L every two days, so $9$ L last $12$ days (twelve servings of $0.75$ : $12 \times 0.75 = 9$ ).

Exercise 30.10 ★★

Milo computes $5.6 \times 3 = 15.18$ , reasoning “ $5 \times 3 = 15$ and $6 \times 3 = 18$ ”. Use an estimate to show the answer must be wrong, then compute correctly.

Solution

Solution of Exercise 30.10.

Estimate: $5.6$ is more than $5.5$ , and $5.5 \times 3 = 16.5$ — so the product must be more than $16.5$ , and $15.18$ is too small. Correctly: $56 \times 3 = 168$ , one decimal digit, so $5.6 \times 3 = 16.8$ . Milo glued his two partial products ( $15$ and $18$ ) side by side instead of adding them with their places: $15 + 1.8 = 16.8$ .

Exercise 30.11 ★★

A runner’s four laps took $65.4$ s, $64.9$ s, $65.0$ s and $66.1$ s.

Compute the total time of the four laps.
The record for four laps is $260$ s. Did she beat it? By how much?

Solution

Solution of Exercise 30.11.

1. $65.4 + 64.9 + 65.0 + 66.1 = 261.4$ s.

2. $261.4 > 260$ : she did not beat the record; she was $1.4$ s over it.