Negative numbers, factors, multiples and primes – Year 7 maths test answers to download as PDF

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1. Answer to Question 1 (4 marks)

   1. Start at \(-7\) on the number line and move 12 places to the right. After 7 steps you reach 0, and 5 more steps take you to 5. (1 mark)
   2. Subtracting a negative number is the same as adding the positive number: \(-5-(-9)=-5+9=\)4. (1 mark)
   3. Multiply the numbers, \(6\times7=42\). The signs are different (one negative, one positive), so the answer is negative: \(-42\). (1 mark)
   4. Divide the numbers, \(56\div8=7\). Both numbers are negative, and two like signs give a positive answer: 7. (1 mark)

   Check (d) by multiplying back: \(7\times(-8)=-56\). Check (b) on a thermometer: rising from \(-5\) °C by 9 degrees gives 4 °C.

2. Answer to Question 2 (4 marks)

   1. Find the factors in pairs that multiply to 30: \(1\times30\), \(2\times15\), \(3\times10\), \(5\times6\). The factors are 1, 2, 3, 5, 6, 10, 15, 30. Award 1 mark for at least six correct and 2 marks for all eight with no extras.
   2. The factors of 45 are 1, 3, 5, 9, 15, 45. The numbers in both lists are 1, 3, 5 and 15, so the highest common factor is 15. (1 mark)
   3. List multiples until one appears in both lists. Multiples of 6: 6, 12, 18, 24. Multiples of 8: 8, 16, 24. The lowest common multiple is 24. (1 mark)

   Check (b): \(30=15\times2\) and \(45=15\times3\), and 2 and 3 share no further factor.

3. Answer to Question 3 (4 marks)

   1. Divide by primes repeatedly: \(72\div2=36\), \(36\div2=18\), \(18\div2=9\), \(9\div3=3\), \(3\div3=1\). So \(72=2\times2\times2\times3\times3\) (1 mark for a correct tree or ladder). In index form this is \(2^{3}\times3^{2}\) (1 mark).
   2. Test each number from 21 to 39. Even numbers and multiples of 3, 5 or 7 are ruled out. The primes left are 23, 29, 31, 37. (1 mark)
   3. A prime has exactly two factors, 1 and itself. The digits of 57 add to 12, so 57 divides by 3: \(57=3\times19\). It has factors 3 and 19 as well as 1 and 57, so it is not prime. (1 mark)

   Check (a): \(8\times9=72\).

4. Answer to Question 4 (4 marks)

   1. The coldest town has the lowest temperature. The most negative value is \(-9\) °C, so Inverness was coldest. (1 mark)
   2. The warmest town is Cardiff at 2 °C. The difference is \(2-(-9)=2+9=\)11 °C. On a thermometer you count 9 degrees up to zero and 2 more above it. (1 mark)
   3. Leeds starts at \(-1\) °C and falls by 6 degrees, so we subtract: \(-1-6=\)\(-7\) °C. (1 mark)
   4. Aberdeen starts at \(-4\) °C and rises by 7 degrees: \(-4+7=3\) °C. Since \(3\gt0\), the temperature is above freezing point. (1 mark)

   Ordered from coldest to warmest the towns are Inverness, Aberdeen, Leeds, Belfast, Cardiff.

5. Answer to Question 5 (4 marks)

   1. List the multiples of each number. Multiples of 12: 12, 24, 36, 48. Multiples of 18: 18, 36, 54 (1 mark). The first number in both lists is 36, so the LCM is 36 (1 mark).
   2. Red finishes a lap at 12, 24, 36… seconds and blue at 18, 36… seconds. Both teams are next on the line together at the LCM, after 36 seconds (1 mark). By then red has run \(36\div12=3\) laps and blue has run \(36\div18=2\) laps (1 mark).

   Prime factors give the same answer: \(12=2^{2}\times3\) and \(18=2\times3^{2}\), so the LCM is \(2^{2}\times3^{2}=36\). After 72 seconds they meet again, with 6 and 4 laps.

Total : 20 marks