AQA GCSE Maths · 3.1.1

Structure and calculation

50 maths questions with worked answers with answers for 3.1.1 Structure and calculation in AQA GCSE Maths (spec 8300). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official AQA material.

All 50 questions and answers

Question 1 · Order · N1
Put −7, 4, −2 and 0 in ascending order.
Answer:
Ascending means smallest first.
On a number line, −7 < −2 < 0 < 4.
Answer: −7, −2, 0, 4.
Question 2 · Order · N1
Which is greater: −0.3 or −0.35?
Answer:
Write −0.3 as −0.30.
−0.30 > −0.35, so −0.3 is greater.
Question 3 · Order · N1
Insert < or >: −6 __ −9.
Answer:
−6 is closer to zero and lies to the right of −9.
Therefore −6 > −9.
Question 4 · Place value · N2
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 5 · Place value · N2
Calculate 4.8 + 0.37.
Answer:
Align decimal places: 4.8 = 4.80.
Add: 4.80 + 0.37 = 5.17.
Question 6 · Integers · N2
Calculate −8 + 13.
Answer:
Add 8 to −8 to reach 0, then add the remaining 5.
−8 + 13 = 5.
Question 7 · Integers · N2
Calculate 6 − (−9).
Answer:
Subtracting a negative is the same as adding.
6 − (−9) = 6 + 9 = 15.
Question 8 · Integers · N2
Calculate (−4) × (−7).
Answer:
A negative multiplied by a negative is positive.
4 × 7 = 28, so (−4) × (−7) = 28.
Question 9 · Integers · N2
Calculate 45 ÷ (−5).
Answer:
A positive divided by a negative is negative.
45 ÷ 5 = 9, so 45 ÷ (−5) = −9.
Question 10 · Order of operations · N3
Calculate 7 + 3 × 4.
Answer:
Multiply first: 3 × 4 = 12.
Then 7 + 12 = 19.
Question 11 · Order of operations · N3
Calculate (18 − 6) ÷ 3.
Answer:
Work inside the brackets: 18 − 6 = 12.
Then 12 ÷ 3 = 4.
Question 12 · Order of operations · N3
Calculate 2³ + 5 × 2.
Answer:
Evaluate the power and product: 2³ = 8 and 5 × 2 = 10.
Then 8 + 10 = 18.
Question 13 · Order of operations · N3
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 14 · Inverse · N3
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 15 · Reciprocals · N3
What is the reciprocal of 58?
Answer:
Swap numerator and denominator.
The reciprocal of 58 is 85.
Question 16 · Reciprocals · N3
Calculate the reciprocal of −4.
Answer:
The reciprocal is 1 divided by the number.
1 ÷ (−4) = −14.
Question 17 · Factors · N4
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 18 · Factors · N4
Find the highest common factor of 24 and 36.
Answer:
24 = 2³ × 3 and 36 = 2² × 3².
Use the shared factors: 2² × 3 = 12.
Question 19 · Multiples · N4
Find the lowest common multiple of 6 and 8.
Answer:
Multiples of 6: 6, 12, 18, 24.
Multiples of 8: 8, 16, 24.
The first common multiple is 24.
Question 20 · Prime numbers · N4
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 21 · Prime factorisation · N4
Write 84 as a product of prime factors in index form.
Answer:
84 = 2 × 42 = 2 × 2 × 21.
21 = 3 × 7, so 84 = 2² × 3 × 7.
Question 22 · Prime factorisation · N4
Write 180 as a product of prime factors in index form.
Answer:
180 = 18 × 10 = 2 × 3 × 3 × 2 × 5.
Collect equal factors: 2² × 3² × 5.
Question 23 · Counting · N5
List all outcomes when a coin is tossed twice, using H and T.
Answer:
First toss H gives HH or HT.
First toss T gives TH or TT.
Outcomes: HH, HT, TH, TT.
Question 24 · Counting · N5
A café offers 3 soups and 4 sandwiches. How many soup-and-sandwich choices are possible?
Answer:
Each of 3 soups can be paired with 4 sandwiches.
3 × 4 = 12 choices.
Question 25 · Counting · N5 · Higher only
A code has 4 choices for each of 3 positions, with repetition allowed. How many codes are possible?
Answer:
There are 4 choices for each of 3 positions.
4 × 4 × 4 = 4³ = 64 codes.
Question 26 · Powers · N6
Calculate 7².
Answer:
7² means 7 × 7.
7 × 7 = 49.
Question 27 · Powers · N6
Calculate 3³.
Answer:
3³ means 3 × 3 × 3.
3 × 3 × 3 = 27.
Question 28 · Roots · N6
Find the positive square root of 144.
Answer:
12 × 12 = 144.
The positive square root is 12.
Question 29 · Roots · N6
Calculate the cube root of 125.
Answer:
5 × 5 × 5 = 125.
The cube root is 5.
Question 30 · Roots · N6 · Higher only
Between which two consecutive integers is √50?
Answer:
7² = 49 and 8² = 64.
49 < 50 < 64, so 7 < √50 < 8.
Question 31 · Indices · N7
Simplify 3⁴ × 3² as a power of 3.
Answer:
For multiplication with the same base, add the indices.
3⁴ × 3² = 3⁶.
Question 32 · Indices · N7
Simplify 5⁷ ÷ 5³ as a power of 5.
Answer:
For division with the same base, subtract the indices.
5⁷ ÷ 5³ = 5⁴.
Question 33 · Indices · N7
Calculate 2⁻³.
Answer:
A negative index means take the reciprocal.
2⁻³ = 12³ = 18.
Question 34 · Indices · N7
Calculate 9⁰.
Answer:
Any non-zero number to the power zero equals 1.
Therefore 9⁰ = 1.
Question 35 · Indices · N7 · Higher only
Calculate 81¹².
Answer:
A power of one half means the square root.
√81 = 9.
Question 36 · Indices · N7 · Higher only
Calculate 27²³.
Answer:
The denominator 3 means cube root; the numerator 2 means square.
The cube root of 27 is 3, and 3² = 9.
Question 37 · Indices · N7 · Higher only
Calculate 16⁻¹².
Answer:
A negative power gives a reciprocal; a power of one half gives a square root.
√16 = 4, so the value is 14.
Question 38 · Surds · N8 · Higher only
Simplify √72.
Answer:
72 = 36 × 2.
√72 = √36 × √2 = 6√2.
Question 39 · Surds · N8 · Higher only
Simplify 3√5 + 2√5.
Answer:
The surds have the same root, so add the coefficients.
3√5 + 2√5 = (3 + 2)√5 = 5√5.
Question 40 · Surds · N8 · Higher only
Simplify √3 × √12.
Answer:
Combine the roots: √3 × √12 = √36.
√36 = 6.
Question 41 · Surds · N8 · Higher only
Rationalise 4√2.
Answer:
Multiply numerator and denominator by √2.
4√2 = 4√22 = 2√2.
Question 42 · Exact values · N8
A circle has radius 3 cm. Give its exact circumference.
Answer:
Circumference = 2πr.
Substitute r = 3: 2π × 3 = 6π cm.
Question 43 · Standard form · N9
Write 560,000 in standard form.
Answer:
Move the decimal point 5 places left: 560,000 = 5.6 × 100,000.
Therefore 560,000 = 5.6 × 10⁵.
Question 44 · Standard form · N9
Write 0.000072 in standard form.
Answer:
Move the decimal point 5 places right to get 7.2.
Therefore 0.000072 = 7.2 × 10⁻⁵.
Question 45 · Standard form · N9
Write 4.3 × 10³ as an ordinary number.
Answer:
10³ = 1,000.
4.3 × 1,000 = 4,300.
Question 46 · Standard form · N9
Calculate (2 × 10³)(3 × 10⁴) in standard form.
Answer:
Multiply the numbers and add the indices.
(2 × 3) × 10⁷ = 6 × 10⁷.
Question 47 · Standard form · N9
Calculate (8 × 10⁶) ÷ (2 × 10²) in standard form.
Answer:
Divide the numbers and subtract the indices.
(8 ÷ 2) × 10⁴ = 4 × 10⁴.
Question 48 · Standard form · N9
Order from smallest to largest: 9 × 10², 2 × 10³, 1.1 × 10³.
Answer:
Convert to ordinary numbers: 900, 2,000 and 1,100.
Smallest first: 9 × 10², 1.1 × 10³, 2 × 10³.
Question 49 · Counterexample · N4
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 50 · Error check · N3
A student says −3² = 9. What is the correct value?
Answer:
The power is evaluated before the negative sign.
−3² = −(3²) = −9.

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