---
title: "Computing with Decimals"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 30
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/30-computing-with-decimals
---

# Chapter 30 — Computing 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](https://one-course.com/books/math/1/en/chapter/38-decimal-numbers#ch-g6-decimals).)

## 30.1 Adding and subtracting

**Method 30.1 (Columns with a point).**

1. Write the numbers with the *[decimal points](https://one-course.com/books/math/1/en/chapter/25-tenths-and-hundredths#def-g4-decimals-point) aligned* (then units are under units, tenths under tenths);
2. pad the shorter decimal part with [zeros](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) ;
3. add or subtract as with whole numbers;
4. 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.](https://one-course.com/images/onecourse/chapters/math-1/g5-decimalops/fig-c592e82b707c.svg)

*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](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) 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 $1.20$:

1. notebooks: $2.45 \times 3 = 7.35$ ;
2. total: $7.35 + 1.20 = 8.55$ ;
3. 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 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 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](#ex-g5-decimalops-complement)): $6.2$; $3.75$; $9.99$.

**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 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 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 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 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 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 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 of Exercise 30.10.**

Estimate: $5.6$ is more than $5.5$, and $5.5 \times 3 = 16.5$ — so the [product](https://one-course.com/books/math/1/en/chapter/10-multiplication-first-steps#def-g2-mult-def) 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](https://one-course.com/books/math/1/en/chapter/10-multiplication-first-steps#def-g2-mult-def) ($15$ 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.

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

**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.