Skip to content
Marlbridge
Free trial

Flashcards · 9260 · Topic 1

Number flashcards

37 cards on Topic 1, subtopics 3.1.1 to 3.1.3. Pick a subtopic or study them all.

Study mode needs JavaScript. Every card is listed below, so you can still cover the answers and test yourself.

All 37 cards

For a quick read-through or printing. Cover the right-hand side to test yourself. Cards marked Extension are Extension content only.

3.1.1 Structure and calculation

Prime number
A number with exactly two factors, 1 and itself. 1 is not prime; the only even prime is 2. Primes up to 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Find the HCF and LCM of 84 and 90
84 = 2² × 3 × 7 and 90 = 2 × 3² × 5. HCF (shared primes, lower powers) = 2 × 3 = 6. LCM (every prime, higher powers) = 2² × 3² × 5 × 7 = 1260. Check: 6 × 1260 = 84 × 90.
A lamp flashes every 14 s and a buzzer every 20 s. They start together. When do they next coincide?
"Next together" means LCM. 14 = 2 × 7, 20 = 2² × 5. LCM = 2² × 5 × 7 = 140 seconds (2 minutes 20 seconds).
Work out 20 − 2 × 3², and compare −2² with (−2)²
Indices first: 3² = 9, then 2 × 9 = 18, then 20 − 18 = 2. −2² = −4 (square, then negate). (−2)² = 4.
Roots of negative numbers: ∛(−125) and √(−25)
∛(−125) = −5, because (−5)³ = −125. Odd roots of negatives exist. √(−25) is not a real number. Even roots of negatives do not exist.
Simplify 4a³ × 6a⁻⁷, with a positive index
Multiply the numbers: 4 × 6 = 24. Add the indices: 3 + (−7) = −4. 24a⁻⁴ = 24/a⁴
Evaluate 32^(2/5) and 4^(−3/2)Extension
32^(2/5): fifth root of 32 is 2, and 2² = 4. 4^(−3/2) = 1/4^(3/2) = 1/(√4)³ = 1/8. Root first, then power; a negative power is a reciprocal.
Simplify √45 + √20Extension
Take out the largest square factor: √45 = 3√5 and √20 = 2√5. 3√5 + 2√5 = 5√5
Rationalise 8/(4 + √3)Extension
Multiply top and bottom by the conjugate 4 − √3. Denominator: 16 − 3 = 13. 8(4 − √3)/13 = (32 − 8√3)/13
Calculate (6 × 10⁴) × (5 × 10⁻⁷) in standard form
6 × 5 = 30 and 10⁴ × 10⁻⁷ = 10⁻³, giving 30 × 10⁻³. Tidy so the first number is from 1 up to 10: 3 × 10⁻².
30 pupils study Spanish or French. 18 study Spanish and 16 study French. How many study both?
n(A ∪ B) = n(A) + n(B) − n(A ∩ B), so 30 = 18 + 16 − both. Both = 34 − 30 = 4.
Error interval
The range of true values that round to a given value: value ± half the rounding unit, with ≤ at the bottom and < at the top. Example: 530 to the nearest 10 gives 525 ≤ x < 535.
Round 0.003951 to 2 significant figures
The first significant figures are 3 and 9; the next digit is 5, so round up: 0.0040. The trailing zero shows the accuracy and must be written.
Estimate 9.7 × 0.52 ÷ 4.9
Round each number to 1 significant figure and show it: 10 × 0.5 ÷ 5 = 1.
Bounds: how do you find the upper bound of a − b and of a ÷ b?Extension
Use the upper bound of a with the lower bound of b in both. Example: 155 ÷ 0.55 is the upper bound of p ÷ q when p = 150 (nearest 10) and q = 0.6 (1 d.p.).

3.1.2 Fractions, decimal and percentages

Which fractions give terminating decimals?
A fraction in lowest terms terminates if its denominator has only 2 and 5 as prime factors (5/16 = 0.3125). Otherwise it recurs (1/6 = 0.16̇).
Write 0.1̇2̇ (0.1212…) as a fractionExtension
x = 0.1212…, 100x = 12.1212… Subtract: 99x = 12, so x = 12/99 = 4/33.
Write 0.4̇5̇ (0.4545…) as a fractionExtension
99x = 45, so x = 45/99 = 5/11. Two repeating digits give a denominator of 99.
Find 3/7 of 91
91 ÷ 7 × 3 = 13 × 3 = 39.
Write 85 cm as a percentage of 2.5 m
Same units first: 2.5 m = 250 cm. 85/250 = 17/50 = 34%.
Multiplier for an increase or decrease of r%
Increase: 1 + r/100 (8% up → × 1.08). Decrease: 1 − r/100 (8% down → × 0.92). New amount = original × multiplier.
A bike costs £70 in a shop and £91 online. Find the percentage change.
(change ÷ original) × 100 = (91 − 70)/70 × 100 = 30% increase.
Simple interest on £1200 at 4% a year for 3 years
Simple interest = P × r/100 × t = 1200 × 0.04 × 3 = £144. It is the same every year, unlike compound interest.
Simple and compound interest: the difference
Simple interest uses the original sum every year. Compound interest is worked out on the balance at the start of each year, so the interest grows. They are equal after one year.
After a 16% increase a price is £1566. Find the original price.Extension
Divide by the multiplier: 1566 ÷ 1.16 = £1350. Check: 1350 × 1.16 = 1566.
Compound interest formulaExtension
Value = P(1 + r/100)ⁿ P = amount invested, r = rate (%), n = years. Example: £2000 at 5% for 4 years = 2000 × 1.05⁴ = £2431.01.

3.1.3 Ratio and proportion

Write 1.5 m : 60 cm in simplest form
Same unit first: 150 : 60. Divide by 30: 5 : 2. Leave the units off.
Share 1350 in the ratio 4 : 5 : 6
Parts = 15; one part = 1350 ÷ 15 = 90. Shares: 360, 450, 540 (they add to 1350).
Boys to girls is 7 : 5 and there are 45 girls. How many boys?
One part = 45 ÷ 5 = 9. Boys = 7 × 9 = 63.
Combine P : Q = 3 : 5 and Q : R = 10 : 7
Make Q the same in both: P : Q = 6 : 10. P : Q : R = 6 : 10 : 7
Best buy: 3 kg for 7.50 or 5 kg for 12.00?
Cost per kg: 7.50 ÷ 3 = 2.50; 12.00 ÷ 5 = 2.40. The 5 kg pack is the better buy.
How can you test a table for direct or inverse proportion?
Direct: y ÷ x is the same for every pair (graph is a straight line through the origin). Inverse: x × y is the same for every pair.
4 machines take 18 hours to do a job. How long do 6 machines take?
Inverse proportion: total = 4 × 18 = 72 machine-hours. 72 ÷ 6 = 12 hours (assuming every machine works at the same rate).
A value rises by 10% then falls by 10%. What is the overall change?
Multiply the multipliers: 1.1 × 0.9 = 0.99, a 1% decrease, not zero.
Exponential growth and decay modelExtension
Amount = start × kᵗ, with k = 1 + r/100 for growth or k = 1 − r/100 for decay. Example: 6400 litres losing 10% a day: 6400 × 0.9³ = 4665.6 litres after 3 days.
A 250 g sample loses 20% of its mass each hour. After how many whole hours is it first below 100 g?Extension
Mass = 250 × 0.8ᵗ. Try whole values: t = 4: 102.4 g (not yet). t = 5: 81.92 g. Answer: 5 hours.
A train covers 210 km in 2 h 20 min. Find its speed.
2 h 20 min = 2⅓ h = 7/3 h (not 2.2 h). 210 ÷ 7/3 = 210 × 3/7 = 90 km/h

Revise Number further