Teach Me Maths

Negative Numbers

Question 11 mark

Here are four numbers: -7,\ -2,\ 1,\ -5. Write down the smallest number.

Hint

Think about which value is furthest left on a number line.

Worked solution

Here are four numbers: -7,\ -2,\ 1,\ -5. Write down the smallest number.

  1. The negatives are less than 1. Of the negatives, -7 is furthest left, so -7 is smallest.

Question 21 mark

Work out -8+5.

Hint

Start at negative eight and move right.

Worked solution

Work out -8+5.

  1. Move 5 places right from -8. You reach -3, so -8+5=-3.

Question 31 mark

Work out 4-9.

Hint

Move left from 4, crossing zero.

Worked solution

Work out 4-9.

  1. Four of the nine moves reach zero. Five more moves left reach -5.

Question 41 mark

Work out -3-4.

Hint

Subtracting a positive number moves left.

Worked solution

Work out -3-4.

  1. Start at -3 and move 4 places left. The result is -7.

Question 51 mark

Work out 6+(-8).

Hint

Adding a negative has the same effect as subtracting its size.

Worked solution

Work out 6+(-8).

  1. 6+(-8)=6-8=-2.

Question 61 mark

Work out -5-(-3).

Hint

Rewrite subtracting a negative as adding a positive.

Worked solution

Work out -5-(-3).

  1. -5-(-3)=-5+3=-2.

Question 71 mark

Work out (-4)\times6.

Hint

Find the size first, then use the signs.

Worked solution

Work out (-4)\times6.

  1. 4\times6=24. The signs are different, so the answer is -24.

Question 81 mark

Work out (-7)\times(-3).

Hint

The two factors have the same sign.

Worked solution

Work out (-7)\times(-3).

  1. 7\times3=21. Two negative factors give a positive product, so the answer is 21.

Question 91 mark

Work out (-42)\div6.

Hint

Use 42 divided by 6, then decide the sign.

Worked solution

Work out (-42)\div6.

  1. 42\div6=7. The signs are different, so (-42)\div6=-7.

Question 101 mark

Work out (-36)\div(-9).

Hint

The signs are the same.

Worked solution

Work out (-36)\div(-9).

  1. 36\div9=4. Dividing a negative by a negative gives a positive answer, so the answer is 4.

Question 112 marks

The temperature is -6^\circC. It rises by 11^\circC, then falls by 8^\circC. Find the final temperature in °C.

Hint

Apply the rise and the fall in order.

Worked solution

The temperature is -6^\circC. It rises by 11^\circC, then falls by 8^\circC. Find the final temperature in °C.

  1. After the rise: -6+11=5^\circC.
  2. After the fall: 5-8=-3^\circC.

Question 122 marks

A temperature rises from -9^\circC to 5^\circC. How many degrees does it rise?

Hint

Count the distance to zero, then from zero to the final temperature.

Worked solution

A temperature rises from -9^\circC to 5^\circC. How many degrees does it rise?

  1. From -9 to 0 is a rise of 9. From 0 to 5 is a rise of 5.
  2. Total rise: 9+5=14^\circC.

Question 132 marks

Find the missing number: -3+\square=8.

Hint

Find how far to move right from negative three to eight.

Worked solution

Find the missing number: -3+\square=8.

  1. It takes 3 to reach zero and another 8 to reach 8.
  2. 3+8=11, so the missing number is 11. Check: -3+11=8.

Question 142 marks

Find the missing number: -6-\square=-10.

Hint

Which positive subtraction moves from negative six to negative ten?

Worked solution

Find the missing number: -6-\square=-10.

  1. Moving from -6 to -10 means moving 4 places left.
  2. So -6-4=-10 and the missing number is 4.

Question 151 mark

Noor says -5+(-2)=7 because two negatives make a positive. Which statement correctly explains the mistake?

Hint

Identify the operation before deciding which sign rule applies.

Worked solution

Noor says -5+(-2)=7 because two negatives make a positive. Which statement correctly explains the mistake?

  1. The sign rule is for multiplication or division; adding -2 moves left from -5, giving -7.
  2. Adding two negatives gives a positive, but the answer should be 10.
  3. The answer should be -3 because the digits must be subtracted.
  4. The answer is 7 because both minus signs can be removed.
  1. The operation is addition, not multiplication. Adding -2 means moving 2 places left from -5.
  2. The correct calculation is -5+(-2)=-7. The first statement explains the error.

Question 16Challenge3 marks

At 6 am the temperature is -8^\circC. It rises, then falls by 7^\circC. The final temperature is -3^\circC. How many degrees was the rise?

Hint

Undo the fall to find the temperature just before it.

Worked solution

At 6 am the temperature is -8^\circC. It rises, then falls by 7^\circC. The final temperature is -3^\circC. How many degrees was the rise?

  1. Before the 7^\circC fall, the temperature was -3+7=4^\circC.
  2. The rise from -8 to 4 is 8+4=12^\circC.
  3. Check: -8+12-7=-3.

Question 17Challenge3 marks

A quiz score starts at -6. Each correct answer adds 5 points and each wrong answer subtracts 3 points. A player answers 7 correctly and 4 wrongly. What is their final score?

Hint

Calculate the points gained and lost, then include the starting score.

Worked solution

A quiz score starts at -6. Each correct answer adds 5 points and each wrong answer subtracts 3 points. A player answers 7 correctly and 4 wrongly. What is their final score?

  1. Points gained: 7\times5=35. Points lost: 4\times3=12.
  2. Final score: -6+35-12=29-12=17.

Question 18Challenge3 marks

Two integers have a sum of -17. One of the integers is -11.

(a)

What is the other integer?

2 marks

(b)

How much greater is the larger integer than the smaller integer?

1 mark

Hint

Think about what you must add to -11 to reach -17.

Worked solution

Two integers have a sum of -17. One of the integers is -11.

Part (a)

What is the other integer?

  1. The other integer is -17-(-11)=-17+11=-6.
  2. Check: -11+(-6)=-17.

Part (b)

How much greater is the larger integer than the smaller integer?

  1. -6 is further right on the number line than -11, so -6 is the larger integer.
  2. -6-(-11)=-6+11=5, so it is 5 greater.

Question 19Challenge3 marks

Two numbers A and B satisfy A\times(-6)=36 and B-A=-11. Work out B.

Hint

Find A from the first calculation, then use it in the second. Keep the negative value in brackets.

Worked solution

Two numbers A and B satisfy A\times(-6)=36 and B-A=-11. Work out B.

  1. From A\times(-6)=36, divide by -6: A=36\div(-6)=-6.
  2. Substitute A=-6 into the second calculation: B-(-6)=-11.
  3. Subtracting -6 adds 6, so B+6=-11. Undo the addition: B=-11-6=-17.
  4. Check: (-6)\times(-6)=36 and -17-(-6)=-17+6=-11.

Question 20Challenge4 marks

A freezer starts at -18^\circC. While its door is open, the temperature rises by 3^\circC each minute for 5 minutes. The door then closes. Each complete cooling cycle lowers the temperature by 4^\circC. Work out the fewest complete cooling cycles needed to reach -19^\circC or lower.

Hint

Find the temperature after warming, then the drop needed to reach the target.

Worked solution

A freezer starts at -18^\circC. While its door is open, the temperature rises by 3^\circC each minute for 5 minutes. The door then closes. Each complete cooling cycle lowers the temperature by 4^\circC. Work out the fewest complete cooling cycles needed to reach -19^\circC or lower.

  1. The rise is 5\times3=15^\circC. After warming, the temperature is -18+15=-3^\circC.
  2. The drop from -3^\circC to -19^\circC is 16^\circC.
  3. Each cycle lowers the temperature by 4^\circC, so 16\div4=4 cycles are needed.
  4. Check: three cycles give -3-12=-15^\circC, which is too warm. Four give -3-16=-19^\circC, so 4 is the fewest.

More on this topic: Negative Numbers worksheet with full solutions

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