OCR GCSE Maths · OCR 1

Number Operations and Integers

50 maths questions with worked answers with answers for OCR 1 Number Operations and Integers in OCR GCSE Maths (spec J560). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official OCR material.

All 50 questions and answers

Question 1 · Negative numbers · OCR 1.01a
Calculate −8 × −6.
Answer:
A negative multiplied by a negative is positive.
−8 × −6 = 48.
Question 2 · Negative numbers · OCR 1.01a
Calculate −15 ÷ 3 + 7.
Answer:
Divide first: −15 ÷ 3 = −5.
−5 + 7 = 2.
Question 3 · Place value · OCR 1.01a
Given 23 × 47 = 1081, find 2.3 × 0.47.
Answer:
2.3 is 23 ÷ 10 and 0.47 is 47 ÷ 100.
1081 ÷ 1000 = 1.081.
Question 4 · Negative numbers · OCR 1.01a
The temperature is −6 °C. It rises by 11 °C. What is the new temperature?
Answer:
−6 + 11.
5 °C.
Question 5 · Place value · OCR 1.01a
What is the value of the digit 7 in 38.072?
Answer:
The 7 is in the hundredths column.
Its value is 7100 = 0.07.
Question 6 · Place value · OCR 1.01a
Calculate 4.8 + 0.37.
Answer:
Align decimal places: 4.8 = 4.80.
Add: 4.80 + 0.37 = 5.17.
Question 7 · Integers · OCR 1.01a
Calculate −8 + 13.
Answer:
Add 8 to −8 to reach 0, then add the remaining 5.
−8 + 13 = 5.
Question 8 · Integers · OCR 1.01a
Calculate 6 − (−9).
Answer:
Subtracting a negative is the same as adding.
6 − (−9) = 6 + 9 = 15.
Question 9 · Integers · OCR 1.01a
Calculate (−4) × (−7).
Answer:
A negative multiplied by a negative is positive.
4 × 7 = 28, so (−4) × (−7) = 28.
Question 10 · Integers · OCR 1.01a
Calculate 45 ÷ (−5).
Answer:
A positive divided by a negative is negative.
45 ÷ 5 = 9, so 45 ÷ (−5) = −9.
Question 11 · Error check · OCR 1.01a
A student says 13 + 14 = 27. Correct the calculation.
Answer:
Use denominator 12: 13 = 412 and 14 = 312.
412 + 312 = 712, not 27.
Question 12 · Four rules · OCR 1.01a
Calculate 347 × 26 without a calculator.
Answer:
347 × 20 = 6940 and 347 × 6 = 2082.
6940 + 2082 = 9022.
Question 13 · Four rules · OCR 1.01a
Calculate 1872 ÷ 8 without a calculator.
Answer:
Short division: 18 ÷ 8 = 2 r 2, 27 ÷ 8 = 3 r 3, 32 ÷ 8 = 4.
1872 ÷ 8 = 234.
Question 14 · Four rules · OCR 1.01a
Calculate −12 − 7.
Answer:
Start at −12 and move 7 further down.
−19.
Question 15 · Four rules · OCR 1.01a
Calculate (−3)³.
Answer:
(−3) × (−3) × (−3) = 9 × (−3).
−27.
Question 16 · Four rules · OCR 1.01a
Calculate 5 × (−4) − (−6).
Answer:
5 × (−4) = −20.
−20 + 6 = −14.
Question 17 · HCF and LCM · OCR 1.02a
Buses leave every 12 minutes and trams every 20 minutes. Both leave at 9:00. When do they next leave together?
Answer:
LCM of 12 and 20: 12 = 2² × 3, 20 = 2² × 5, so LCM = 60.
They next leave together at 10:00.
Question 18 · Roots · OCR 1.02a
Find the cube root of 125.
Answer:
5 × 5 × 5 = 125.
∛125 = 5.
Question 19 · Factors · OCR 1.02a
List the positive factors of 18.
Answer:
Use factor pairs: 1 × 18, 2 × 9 and 3 × 6.
Factors: 1, 2, 3, 6, 9, 18.
Question 20 · Roots · OCR 1.02a
Find the positive square root of 144.
Answer:
12 × 12 = 144.
The positive square root is 12.
Question 21 · Roots · OCR 1.02a · Higher only
Between which two consecutive integers is √50?
Answer:
7² = 49 and 8² = 64.
49 < 50 < 64, so 7 < √50 < 8.
Question 22 · Definitions · OCR 1.02a
Write down the multiples of 7 between 30 and 60.
Answer:
7 × 5 = 35 up to 7 × 8 = 56.
35, 42, 49, 56.
Question 23 · Definitions · OCR 1.02a
Find the common factors of 24 and 36.
Answer:
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
Common factors: 1, 2, 3, 4, 6, 12.
Question 24 · Place value · OCR 1.02a
Given 56 × 34 = 1904, find 5.6 × 340.
Answer:
5.6 is 56 ÷ 10 and 340 is 34 × 10.
The answer is the same: 1904.
Question 25 · Primes · OCR 1.02b
List the prime numbers between 20 and 40.
Answer:
A prime has exactly two factors.
23, 29, 31, 37.
Question 26 · Prime factorisation · OCR 1.02b
Write 180 as a product of prime factors in index form.
Answer:
180 = 2 × 90 = 2 × 2 × 45 = 2 × 2 × 3 × 15 = 2 × 2 × 3 × 3 × 5.
180 = 2² × 3² × 5.
Question 27 · Prime factorisation · OCR 1.02b
A = 2³ × 3 × 5² and B = 2 × 3² × 5. Find the HCF of A and B.
Answer:
Take the lowest power of each common prime.
HCF = 2 × 3 × 5 = 30.
Question 28 · Prime numbers · OCR 1.02b
Which number is prime: 21, 29 or 33?
Answer:
21 = 3 × 7 and 33 = 3 × 11, so neither is prime.
29 has only 1 and 29 as positive factors; it is prime.
Question 29 · Counterexample · OCR 1.02b
Give a counterexample to: every odd number is prime.
Answer:
Choose 9: it is odd, but 9 = 3 × 3.
It has a factor other than 1 and itself, so it is not prime.
Question 30 · Prime factors · OCR 1.02b
Write 600 as a product of prime factors in index form.
Answer:
600 = 2 × 300 = 2 × 2 × 150 = 2 × 2 × 2 × 75 = 2³ × 3 × 25.
600 = 2³ × 3 × 5².
Question 31 · Prime factors · OCR 1.02b
Explain why 2 is the only even prime number.
Answer:
Every other even number has 2 as a factor as well as 1 and itself.
So every other even number has more than two factors.
Question 32 · HCF · OCR 1.02c
Find the highest common factor of 84 and 120.
Answer:
84 = 2² × 3 × 7 and 120 = 2³ × 3 × 5.
HCF = 2² × 3 = 12.
Question 33 · LCM · OCR 1.02c
Find the lowest common multiple of 18 and 24.
Answer:
18 = 2 × 3² and 24 = 2³ × 3.
LCM = 2³ × 3² = 72.
Question 34 · HCF and LCM · OCR 1.02c
Use prime factors to find the LCM of 60 and 72.
Answer:
60 = 2² × 3 × 5 and 72 = 2³ × 3².
LCM = 2³ × 3² × 5 = 360.
Question 35 · HCF and LCM · OCR 1.02c
Pencils come in packs of 8 and rubbers in packs of 12. What is the smallest number of each you can buy to have equal numbers?
Answer:
LCM of 8 and 12 = 24.
3 packs of pencils and 2 packs of rubbers.
Question 36 · HCF and LCM · OCR 1.02c
48 apples and 60 oranges are shared into identical bags with nothing left over. What is the greatest number of bags?
Answer:
HCF of 48 and 60: 48 = 2⁴ × 3, 60 = 2² × 3 × 5.
HCF = 2² × 3 = 12 bags.
Question 37 · Order of operations · OCR 1.03a
Calculate 4 + 6 × 3².
Answer:
Index first: 3² = 9. Then multiply: 6 × 9 = 54.
4 + 54 = 58.
Question 38 · Order of operations · OCR 1.03a
Calculate (12 − 4)² ÷ 4.
Answer:
Brackets first: 12 − 4 = 8. Then 8² = 64.
64 ÷ 4 = 16.
Question 39 · Reciprocals · OCR 1.03a
What is the reciprocal of 0.25?
Answer:
0.25 = 14.
The reciprocal is 41 = 4.
Question 40 · Calculator use · OCR 1.03a
Use a calculator to work out √(5.2² + 3.9²). Give your answer to 1 decimal place.
Answer:
5.2² + 3.9² = 27.04 + 15.21 = 42.25.
√42.25 = 6.5.
Question 41 · Order of operations · OCR 1.03a
Calculate 2³ + 5 × 2.
Answer:
Evaluate the power and product: 2³ = 8 and 5 × 2 = 10.
Then 8 + 10 = 18.
Question 42 · Order of operations · OCR 1.03a
Calculate 36 ÷ 3 × 2, working from left to right.
Answer:
Division and multiplication have equal priority; work left to right.
36 ÷ 3 = 12, then 12 × 2 = 24.
Question 43 · Reciprocals · OCR 1.03a
Calculate the reciprocal of −4.
Answer:
The reciprocal is 1 divided by the number.
1 ÷ (−4) = −14.
Question 44 · Error check · OCR 1.03a
A student says −3² = 9. What is the correct value?
Answer:
The power is evaluated before the negative sign.
−3² = −(3²) = −9.
Question 45 · Order of operations · OCR 1.03a
Calculate 20 − 2 × (3 + 4).
Answer:
Brackets: 3 + 4 = 7. Multiply: 2 × 7 = 14.
20 − 14 = 6.
Question 46 · Order of operations · OCR 1.03a
Insert brackets to make 3 + 5 × 4 = 32 true.
Answer:
Add first, then multiply.
(3 + 5) × 4 = 32.
Question 47 · Inverse operations · OCR 1.04a
A number is multiplied by 6 and then 11 is subtracted. The result is 37. Find the number.
Answer:
Work backwards: 37 + 11 = 48.
48 ÷ 6 = 8.
Question 48 · Inverse · OCR 1.04a
A number is multiplied by 6 to give 42. What is the number?
Answer:
Use the inverse operation: 42 ÷ 6 = 7.
Check: 7 × 6 = 42.
Question 49 · Inverse operations · OCR 1.04a
Use the inverse operation to check 952 ÷ 14 = 68.
Answer:
68 × 14 = 680 + 272.
= 952, so the answer is correct.
Question 50 · Inverse operations · OCR 1.04a
The square of a number is 81. What could the number be?
Answer:
The inverse of squaring is square-rooting.
9 or −9.

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