Integers Problems
Integers Problems brings all four operations with positive and negative numbers together in worded questions, the last step of the topic in NSW Year 7 maths (NESA outcome MA4‑INT‑C‑01). The key move is to choose which direction counts as positive, then write a directed number sentence.
You will solve problems about bank balances, profits and losses, golf scores relative to par, temperatures, submarines, mine lifts and time zones, using multiplication for repeated changes, and give each answer in words with its direction. It is part of Year 7 Integers.
Every question with a fully worked solution.
- Integers Problems - Video - Integer Word Problems Watch
Theory
To solve an integer problem, choose which direction is positive, write a directed number sentence, then calculate and answer in words. This Year 7 maths page works through temperatures, walks north and south, submarines, bank balances, quiz scores and time zones.
Step 1. Choose a direction. Decide what is positive, such as north, up, a deposit or a profit. The opposite direction is negative.
Step 2. Write each change as an integer. A drop of \(6^\circ\text{C}\) is \(-6\). A deposit of \(\$20\) is \(+20\).
Step 3. Write a directed number sentence. A change that repeats is a multiplication: \(4\) withdrawals of \(\$15\) is \(4 \times (-15)\).
Step 4. Calculate, then answer in words. Say what the sign means: \(-30\) m means \(30\) m below the surface.
Common words and their signs:
| Positive (+) | Negative (−) |
|---|---|
| above sea level, up, rise | below sea level, down, fall |
| north or east (if chosen) | south or west |
| deposit, profit, earn | withdrawal, loss, debt, fee |
| points won, above par | points lost, below par |
| ahead of UTC | behind UTC |
How to solve an integer problem
- Choose which direction is positive.
- Write each change as an integer, with its sign.
- Write a number sentence; use \(\times\) for a change that repeats.
- Calculate using the order of operations, then answer in words.
| \(11 - (-2)\) | \(=\) | \(11 + 2\) |
| \(\) | \(=\) | \(13\) |
It dropped \(13^\circ\text{C}\).
| \(-3 + 4\) | \(=\) | \(1\) |
North is positive. They are \(1\) km north of Anya’s home, as in the first diagram.
| \(-60 + 25 - 40\) | \(=\) | \(-35 - 40\) |
| \(\) | \(=\) | \(-75\) |
It is at \(-75\) m, which is \(75\) m below the surface.
(i) What was the total profit?
(ii) What was the mean weekly profit?
| \(-130 + 90 + (-40) + 160\) | \(=\) | \(80\) |
| \(80 \div 4\) | \(=\) | \(20\) |
(i) \(\$80\)
(ii) \(\$20\) a week
(i) How many hours is P ahead of Q?
(ii) It is hour \(22\) (10 pm) in P. What time is it in Q?
| \(10 - (-10)\) | \(=\) | \(20\) |
| \(22 - 20\) | \(=\) | \(2\) |
(i) \(20\) hours
(ii) Hour \(2\), which is 2 am
Try a number correct; the rest are wrong. \(5\) correct and \(5\) wrong score \(5 \times 3 = 15\) and \(5 \times (-1) = -5\). \(6\) correct and \(4\) wrong score \(18\) and \(-4\).
| \(15 + (-5)\) | \(=\) | \(10\) |
| \(18 + (-4)\) | \(=\) | \(14\) |
\(5\) correct gives \(10\), too low. \(6\) correct gives \(14\). Zoe got \(6\) correct.
| \(50 + 7 \times (-9)\) | \(=\) | \(50 - 63\) |
| \(\) | \(=\) | \(-13\) |
Her balance is \(-\$13\): she owes \(\$13\).
Common pitfalls
Frequently asked questions
How do you choose which direction is positive?
Pick the natural one: up, north, a deposit or a profit. Then the opposite is negative.
What is a directed number sentence?
A calculation where each change has a sign, such as \(-60 + 25 - 40\).
How do you find how far apart two positions are?
Higher minus lower. From \(-35\) to \(20\) is \(20 - (-35) = 55\).
When do you multiply in an integer problem?
When the same change repeats, such as a fee each week: \(6\) fees of \(\$5\) is \(6 \times (-5)\).
What does a negative answer mean?
It means the opposite of the direction you chose as positive, such as below sea level or money owed.