Year 7 Maths Integers

Dividing Integers

Q: Why does division use the same sign rule as multiplication?

Division undoes multiplication. Since -4 × 3 = -12, -12 ÷ 3 must be -4.

Q: Does a fraction bar mean divide?

Yes. -12/-3 means -12 ÷ (-3), which is 4.

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Theory

To divide two integers, divide their sizes, then use the sign rule: same signs give a positive answer and different signs give a negative answer. This Year 7 maths page shows why division follows the same rule as multiplication, how to divide from left to right, how to find a missing number and how to work out a change per hour or a mean.

Division undoes multiplication. Because $-4 \times 3 = -12$, it follows that $-12 \div 3 = -4$ and $-12 \div (-4) = 3$. So dividing uses the same sign rule as multiplying.

Same signs $\Rightarrow$ positive answer: $-12 \div (-3) = 4$. Different signs $\Rightarrow$ negative answer: $12 \div (-3) = -4$.

A fraction bar also means divide: $\dfrac{-12}{-3} = -12 \div (-3) = 4$.

The mean of a set of values is total $\div$ number of values. The mean of $-5$, $2$ and $-6$ is $-9 \div 3 = -3$.

Split a move of $-12$ into $3$ equal jumps. Each jump is $-4$, so $-12 \div 3 = -4$.

Signs of a quotient (the same as for products):

$\div$	positive	negative
positive	positive	negative
negative	negative	positive

\[ -4 \times 3 = -12 \;\Rightarrow\; -12 \div 3 = -4 \]

- 4 \times 3 = - 12, so - 12 \div 3 = - 4

\[ \dfrac{-12}{-3} = -12 \div (-3) = 4 \]

- 12 / - 3 = 4

\[ \text{mean} = \text{total} \div \text{number of values} \]

mean = total \div number of values

          Work left to right. \(36 \div (-6) \div 2 = -6 \div 2 = -3\). Doing \(-6 \div 2\) first gives \(36 \div (-3) = -12\), which is wrong.
        

          Missing numbers. Undo the division: \(\square \div (-4) = 9\) gives \(\square = 9 \times (-4) = -36\). In \(44 \div \square = -4\) the size is \(11\) and the signs differ, so \(\square = -11\).
        

How to divide integers

Divide the sizes, ignoring the signs.
Same signs: positive answer. Different signs: negative answer.
Several divisions (or a mix of $\times$ and $\div$): work from left to right.

How to find a rate of change

Find the total change, with its sign.
Divide by the number of hours, days or steps.

How to find a mean

Add all the values, keeping the signs.
Divide the total by how many values there are.

Example 1 — Size, then sign

Work out
(i) $-35 \div 7$
(ii) $-48 \div (-6)$
(iii) $\dfrac{-60}{-5}$

Solution

$-35 \div 7$	$=$	$-5$
$-48 \div (-6)$	$=$	$8$
$\dfrac{-60}{-5}$	$=$	$12$

- 35 \div 7 = - 5, - 48 \div (- 6) = 8, - 60 / - 5 = 12

(i) Different signs: $-5$
(ii) Same signs: $8$
(iii) Same signs: $12$

Example 2 — Left to right

Work out $72 \div (-6) \div (-3)$.

Solution

$72 \div (-6) \div (-3)$	$=$	$-12 \div (-3)$
	$=$	$4$

72 \div (- 6) \div (- 3) = - 12 \div (- 3) = 4

$4$

Example 3 — Multiply and divide

Work out $-4 \times 9 \div (-12)$.

Solution

$-4 \times 9 \div (-12)$	$=$	$-36 \div (-12)$
	$=$	$3$

- 4 \times 9 \div (- 12) = - 36 \div (- 12) = 3

Left to right: $3$

Example 4 — How many minutes?

A submarine’s depth changes by $-63$ m. It changes by $-7$ m each minute. For how many minutes does it dive?

Solution

$-63 \div (-7)$

$=$

$9$

- 63 \div (- 7) = 9

Check: $9 \times (-7) = -63$. It dives for $9$ minutes.

Example 5 — Change per hour

It was $4^\circ\text{C}$ at 9 pm and $-16^\circ\text{C}$ at 2 am. The temperature fell at a steady rate. What was the change each hour?

Solution

Total change: $4$ down to $0$, then $16$ more, a fall of $20$. From 9 pm to 2 am is $5$ hours.

$-20 \div 5$

$=$

$-4$

- 20 \div 5 = - 4

The change was $-4^\circ\text{C}$ each hour.

Example 6 — Mean temperature

The minimum temperatures on four nights were $-6^\circ\text{C}$, $3^\circ\text{C}$, $-9^\circ\text{C}$ and $-4^\circ\text{C}$. Find the mean.

Solution

$-6 + 3 + (-9) + (-4)$	$=$	$-16$
$-16 \div 4$	$=$	$-4$

- 6 + 3 + (- 9) + (- 4) = - 16, - 16 \div 4 = - 4

The mean is $-4^\circ\text{C}$.

Example 7 — Missing numbers

Find the missing integer.
(i) $\square \div (-4) = 8$
(ii) $-45 \div \square = 5$

Solution

$8 \times (-4)$	$=$	$-32$
$-45 \div (-9)$	$=$	$5$

8 \times (- 4) = - 32, - 45 \div (- 9) = 5

(i) Undo the division: $-32$
(ii) Size $9$; the answer is positive, so the signs match: $-9$

Common pitfalls

Two negatives give a positive. $-28 \div (-4) = 7$, not $-7$.

One negative gives a negative. $40 \div (-8) = -5$ and $-40 \div 8 = -5$.

Not working left to right. $36 \div (-6) \div 2 = -3$, not $-12$.

Forgetting the sign of a share of a debt. Three people sharing a debt of $\$45$ each owe $-45 \div 3 = -15$ dollars.

Frequently asked questions

What is a negative divided by a negative?

A positive. $-48 \div (-6) = 8$.

What is a positive divided by a negative?

A negative. $12 \div (-3) = -4$.

Why does division use the same sign rule as multiplication?

Division undoes multiplication. Since $-4 \times 3 = -12$, $-12 \div 3$ must be $-4$.

Does a fraction bar mean divide?

Yes. $\dfrac{-12}{-3}$ means $-12 \div (-3)$, which is $4$.

How do you find the mean of some integers?

Add them, keeping the signs, then divide by how many there are. The mean of $-5$, $2$ and $-6$ is $-3$.

\(-35 \div 7\)	\(=\)	\(-5\)
\(-48 \div (-6)\)	\(=\)	\(8\)
\(\dfrac{-60}{-5}\)	\(=\)	\(12\)

\(72 \div (-6) \div (-3)\)	\(=\)	\(-12 \div (-3)\)
\(\)	\(=\)	\(4\)

\(-4 \times 9 \div (-12)\)	\(=\)	\(-36 \div (-12)\)
\(\)	\(=\)	\(3\)

\(-6 + 3 + (-9) + (-4)\)	\(=\)	\(-16\)
\(-16 \div 4\)	\(=\)	\(-4\)

\(8 \times (-4)\)	\(=\)	\(-32\)
\(-45 \div (-9)\)	\(=\)	\(5\)

Dividing Integers

📖 Theory

How to divide integers

How to find a rate of change

How to find a mean

Common pitfalls

Frequently asked questions

Theory