Flashcards · 0580 · Topic 1

Number flashcards

98 cards on Topic 1, subtopics 1.1 to 1.18. Pick a subtopic or study them all.

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1.1 Types of number

Prime number: A number with exactly two factors: 1 and itself. 1 is not prime (it has only one factor). 2 is the only even prime.
Rational and irrational numbers: Rational: can be written as a fraction of two integers, e.g. 0.75 = 3/4, or −5 = −5/1. Irrational: cannot be written as a fraction, e.g. √2, π.
Reciprocal of a number: 1 ÷ the number. The reciprocal of 4 is 1/4; the reciprocal of 2/3 is 3/2. A number multiplied by its reciprocal gives 1.
Finding the HCF from prime factors: Multiply the lowest power of each common prime.
Finding the LCM from prime factors: Multiply the highest power of every prime that appears.
Find the HCF and LCM of 24 and 36 using prime factors: 24 = 2³ × 3 and 36 = 2² × 3² HCF = 2² × 3 = 12 LCM = 2³ × 3² = 72
Factor and multiple: A factor divides exactly into a number: the factors of 12 are 1, 2, 3, 4, 6, 12. A multiple is in its times table: the multiples of 12 are 12, 24, 36, …

1.2 Sets

Set notation: ℰ, n(A) and A′: ℰ: the universal set, everything being considered. n(A): the number of elements in set A. A′: the complement of A, the elements in ℰ that are not in A.
A ∪ B and A ∩ B: A ∪ B (union): elements in A or B or both. A ∩ B (intersection): elements in both A and B.
List the set {x : x is an integer and 2 ≤ x ≤ 6}: {2, 3, 4, 5, 6}. It reads "all x such that x is an integer from 2 to 6".
In a class of 30, 18 study French, 14 study Spanish and 5 study neither. How many study both?: n(F ∪ S) = 30 − 5 = 25 both = 18 + 14 − 25 = 7 Check: 11 French only + 7 Spanish only + 7 both + 5 neither = 30.
How do you fill in a Venn diagram from given totals?: Start with the overlap (intersection), then work outwards to the "only" regions and the region outside the circles. Check that all the regions add up to n(ℰ).
The symbols ∈, ∉, ∅, ⊆ and ⊈Extended: ∈ is an element of; ∉ is not an element of; ∅ the empty set; ⊆ is a subset of; ⊈ is not a subset of. Write the empty set as ∅, not {0} or {∅}.
Three-set Venn diagram: 8 students are in both M and C, and 3 are in all three sets. What goes in the "M and C only" region?Extended: 8 − 3 = 5. The 8 includes the 3 in all three sets, so fill the centre first and subtract.

1.3 Powers and roots

The squares from 11² to 15²: 11² = 121, 12² = 144, 13² = 169, 14² = 196, 15² = 225
The cubes you must recall: 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 10³ = 1000
Work out √144 + ∛64 without a calculator: 12 + 4 = 16
Work out ∛(−27) and (−3)²: ∛(−27) = −3, because (−3)³ = −27. (−3)² = 9. (A negative number has no real square root.)
Work out √(9/25) and 0.2²: √(9/25) = 3/5 0.2² = 0.04 (not 0.4)

1.4 Fractions, decimals and percentages

Proper fraction, improper fraction, mixed number: Proper: numerator smaller than denominator, e.g. 3/8. Improper: numerator larger than denominator, e.g. 11/4. Mixed number: a whole number and a fraction, e.g. 2¾.
Write 3/8 as a decimal and as a percentage: 3 ÷ 8 = 0.375 = 37.5%
Write 35% and 0.64 as fractions in their simplest form: 35% = 35/100 = 7/20 0.64 = 64/100 = 16/25
Convert 3⅖ to an improper fraction, and 11/4 to a mixed number: 3⅖ = (3 × 5 + 2)/5 = 17/5 11/4 = 2 remainder 3 = 2¾
What do 0.2666… and 0.363636… mean in recurring decimal notation?Extended: 0.2666…: only the 6 repeats (one dot, over the 6). 0.363636…: the block 36 repeats (dots over the 3 and the 6).
Write 0.363636… (36 recurring) as a fraction in its simplest formExtended: x = 0.3636…, 100x = 36.3636… 100x − x = 36, so 99x = 36 x = 36/99 = 4/11
Write 0.2666… (6 recurring) as a fraction in its simplest formExtended: 10x = 2.666…, 100x = 26.666… 100x − 10x = 24, so 90x = 24 x = 24/90 = 4/15

1.5 Ordering

The symbols ≠, ≥ and ≤: ≠ not equal to ≥ greater than or equal to ≤ less than or equal to
How do you order numbers given as fractions, decimals and percentages?: Convert them all to decimals, then compare place value by place value (tenths first, then hundredths, …).
Order smallest first: 5/8, 0.6, 62%, 0.59, 2/3: As decimals: 0.625, 0.6, 0.62, 0.59, 0.666… Order: 0.59, 0.6, 62%, 5/8, 2/3
Which is larger: 0.3 or 0.25? And −7 or −4?: 0.3 is larger (3 tenths beats 2 tenths). −4 is larger: the negative number further from zero is smaller, so −7 < −4.

1.6 The four operations

Order of operations (BIDMAS): Brackets, Indices, then Division and Multiplication (left to right), then Addition and Subtraction (left to right). 28 − 8 ÷ 2 = 28 − 4 = 24, not 10.
How do you divide by a fraction? Work out 3/4 ÷ 2/5: Multiply by the reciprocal of the second fraction. 3/4 × 5/2 = 15/8 = 1⅞
Work out 2/3 + 1/4: Use a common denominator: 8/12 + 3/12 = 11/12
Work out 1½ × 2⅔: Change mixed numbers to improper fractions first: 3/2 × 8/3 = 24/6 = 4
The temperature is −4 °C and rises by 9 °C. What is the new temperature? Also work out (−6) × (−3).: −4 + 9 = 5 °C (−6) × (−3) = 18 (negative × negative = positive)

1.7 Indices I

Rules of indices for multiplying and dividing powers of the same base: aᵐ × aⁿ = aᵐ⁺ⁿ (add the indices) aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract the indices)
Rule for a power of a power: (aᵐ)ⁿ = aᵐⁿ (multiply the indices). For example (5²)³ = 5⁶.
Zero and negative indices: work out 7⁰ and 2⁻³: a⁰ = 1 (a ≠ 0), so 7⁰ = 1. a⁻ⁿ = 1/aⁿ, so 2⁻³ = 1/2³ = 1/8.
Fractional indices: what do a^(1/n) and a^(m/n) mean?Extended: a^(1/n) = ⁿ√a (the nth root). a^(m/n) = (ⁿ√a)ᵐ: take the root first, then the power.
Work out 8^(2/3) and 16^(−1/2)Extended: 8^(2/3) = (∛8)² = 2² = 4 16^(−1/2) = 1/√16 = 1/4

1.8 Standard form

Standard form: A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.
Write 4 560 000 and 0.00032 in standard form: 4 560 000 = 4.56 × 10⁶ 0.00032 = 3.2 × 10⁻⁴
Work out (6 × 10⁴) × (4 × 10³), giving the answer in standard form: 6 × 4 = 24 and 10⁴ × 10³ = 10⁷, giving 24 × 10⁷. Re-normalise: 2.4 × 10⁸
Exam trap: an answer of 42 × 10⁵: Not standard form, because 42 is not between 1 and 10. It is 4.2 × 10⁶. Always re-check A after multiplying or dividing.

1.9 Estimation

Decimal places or significant figures: where do you start counting?: Decimal places: from the decimal point. Significant figures: from the first non-zero digit. 0.006382 is 0.01 to 2 d.p. but 0.0064 to 2 s.f.
How do you estimate a calculation?: 1. Round every number to 1 significant figure. 2. Write the rounded calculation down (this earns the method mark). 3. Work it out mentally.
Estimate (38.6 × 5.12) ÷ 0.207: (40 × 5) ÷ 0.2 = 200 ÷ 0.2 = 1000
Round 8.0449 to 1 decimal place: 8.0. Look only at the next digit (4). Do not "double round" 8.0449 → 8.045 → 8.05 → 8.1.
Round 299 600 to 3 s.f., and 7.4951 to 2 d.p.: 299 600 → 300 000 (the 9s carry) 7.4951 → 7.50 (keep the 0 to show 2 d.p.)
230 eggs are packed in boxes of 6. How many boxes are needed, and how many boxes can be filled?: 230 ÷ 6 = 38.33… Boxes needed to hold them all: 39 (round up). Full boxes: 38 (round down).

1.10 Limits of accuracy

Bounds of a value rounded to the nearest unit u: The value lies within half a unit either side: from (value − u/2) up to, but not including, (value + u/2).
A mass is 4.6 kg to 1 decimal place. Write the error interval.: 4.55 ≤ m < 4.65
A length is 30 cm to the nearest 10 cm. State its lower and upper bounds.: Lower bound 25 cm, upper bound 35 cm: 25 ≤ l < 35.
Why is the upper bound not included in the error interval (< not ≤)?: A value exactly at the upper bound rounds up to the next value: 4.65 kg to 1 d.p. is 4.7 kg, not 4.6 kg.
Bounds of a calculation: the combinations that give the MAXIMUM (positive quantities)Extended: Add: UB + UB Subtract: UB − LB Multiply: UB × UB Divide: UB ÷ LB Subtraction and division cross over.
Bounds of a calculation: the combinations that give the MINIMUM (positive quantities)Extended: Add: LB + LB Subtract: LB − UB Multiply: LB × LB Divide: LB ÷ UB
a = 8 and b = 3, both to the nearest whole number. Find the upper bound of a − b.Extended: UB of a − LB of b = 8.5 − 2.5 = 6

1.11 Ratio and proportion

How do you share an amount in a given ratio?: Add the parts, divide the total by the number of parts, then multiply by each share. Share 180 in the ratio 2 : 3 : 4: 9 parts, 180 ÷ 9 = 20, giving 40, 60 and 80.
Simplify the ratio 1.5 kg : 500 g: Use the same units first: 1500 g : 500 g = 3 : 1
Two ages are in the ratio 3 : 5 and differ by 12 years. Find the ages.: The difference is 5 − 3 = 2 parts = 12, so 1 part = 6. Ages: 18 and 30.
5 pens cost $3.40. How much do 8 pens cost?: Unitary method: one pen costs 3.40 ÷ 5 = $0.68, so 8 pens cost 8 × 0.68 = $5.44.
Money is shared in the ratio 2 : 3. What fraction of the total is the first share?: 2/5 (2 parts out of 2 + 3 = 5 parts).

1.12 Rates

Average speed (a formula you must know): average speed = total distance ÷ total time Also: distance = speed × time, time = distance ÷ speed.
Convert 72 km/h to m/s: 72 km = 72 000 m and 1 h = 3600 s. 72 000 ÷ 3600 = 20 m/s
A bus travels 54 km in 1 hour 20 minutes. Find its average speed.: 20 minutes = 20/60 = ⅓ hour (not 0.2 hours), so the time is 4/3 hours. 54 ÷ 4/3 = 40.5 km/h
A car drives 60 km at 30 km/h, then 60 km at 60 km/h. What is the average speed for the whole journey?: Time: 2 h + 1 h = 3 h. Average speed = 120 ÷ 3 = 40 km/h, not the mean of the two speeds (45 km/h).
Density, pressure and population density: density = mass ÷ volume (e.g. g/cm³) pressure = force ÷ area (e.g. N/m²) population density = population ÷ area The syllabus says these formulas are given in the question.
A metal block has mass 540 g and volume 200 cm³. Find its density.: 540 ÷ 200 = 2.7 g/cm³

1.13 Percentages

Percentage change: percentage change = (change ÷ original value) × 100 Divide by the ORIGINAL value, not the new one.
Multipliers for a 20% increase and a 20% decrease: Increase by 20%: × 1.2 Decrease by 20%: × 0.8
Write 18 as a percentage of 24: 18 ÷ 24 × 100 = 75%
Simple and compound interest on $500 at 4% per year for 3 years: Simple (on the original amount each year): 500 × 0.04 × 3 = $60 interest. Compound (on the running total): 500 × 1.04³ = $562.43, so $62.43 interest.
A price rises by 20%, then falls by 20%. Is it back to the original price?: No. 1.2 × 0.8 = 0.96, so it is 4% lower than the original.
Reverse percentage: a price is $84 after a 20% increase. Find the original price.Extended: Divide by the multiplier: 84 ÷ 1.2 = $70. (Not 84 × 0.8 = $67.20.)

1.14 Using a calculator

When should you round in a calculator question?: Only at the end. Keep full values with the Ans or memory key. If a later part uses an earlier answer, use the unrounded value.
Accuracy for answers on the calculator papers, unless the question says otherwise: Non-exact answers to 3 significant figures; angles in degrees to 1 decimal place. Use the π key or 3.142.
How do you enter (3.2 + 4.7) ÷ (2.1 × 0.6) correctly?: Put brackets round the whole numerator and the whole denominator. Without them the calculator divides only part of the expression.
Why does typing −4² give −16?: The calculator squares 4 first, then applies the minus sign. To square −4, type (−4)², which is 16.
How do you enter 3 hours 20 minutes, and read 5.15 hours?: Enter 3 + 20 ÷ 60 (= 3.333… h), never 3.20. 5.15 h: 0.15 × 60 = 9, so 5 hours 9 minutes (not 5 h 15 min).
The display shows 2.3E−05. How do you write the value?: 2.3 × 10⁻⁵ = 0.000023. Never copy "E−05" as the answer.

1.15 Time

Write 3.15 p.m., 12.30 a.m. and 12.30 p.m. in the 24-hour clock: 3.15 p.m. → 15 15 12.30 a.m. (just after midnight) → 00 30 12.30 p.m. (just after noon) → 12 30
How long is it from 09 47 to 13 22?: Count on: 09 47 → 10 00 is 13 min; 10 00 → 13 00 is 3 h; 13 00 → 13 22 is 22 min. Total 3 h 35 min.
Exam trap: finding a time interval by working out 13.22 − 9.47: Times are not decimals: there are 60 minutes in an hour, not 100. Count on to the next hour instead.
City Q is 5 hours ahead of City P. A flight leaves P at 21 40 (P time) and takes 8 h 45 min. What is the local time in Q when it lands?: 21 40 + 8 h 45 min = 06 25 the next day, P time. Add 5 hours: 11 25 the next day, Q time.

1.16 Money

A calculator shows 4.8 for an amount in dollars. How do you write it?: $4.80. Money is given to the nearest cent (2 decimal places).
Exchange rate 1 unit of A = r units of B. How do you convert?: A to B: multiply by r. B to A: divide by r.
£1 = $1.28. Convert £250 to dollars and $500 to pounds.: £250 × 1.28 = $320 $500 ÷ 1.28 = 390.625 = £390.63
How can you check a currency conversion?: Sense check which amount should be larger. If £1 buys more than $1, the dollar amount must be the larger number.

1.17 Exponential growth and decay

Exponential growth and decay formulaExtended: value after n periods = starting value × (multiplier)ⁿ Growth by r%: multiplier = 1 + r/100 Decay by r%: multiplier = 1 − r/100
800 bacteria increase by 15% each hour. How many are there after 5 hours?Extended: 800 × 1.15⁵ = 1609.08…, so about 1609 bacteria.
A car worth $12 000 loses 18% of its value each year. Find its value after 3 years.Extended: 12 000 × 0.82³ = $6616.42 (to the nearest cent).
Exam trap: depreciation by 18% each yearExtended: Do not subtract 18% of the ORIGINAL value each year: that is a linear (simple) decrease. Multiply by 0.82 once for each year.
$1000 is invested at 5% compound interest per year. After how many whole years is it first worth more than $1200?Extended: Try values of n: 1000 × 1.05³ = $1157.63; 1000 × 1.05⁴ = $1215.51. After 4 years.

1.18 Surds

How do you simplify a surd such as √48?Extended: Use √(ab) = √a × √b with the largest square factor: √48 = √(16 × 3) = 4√3.
Simplify √75 − √12Extended: 5√3 − 2√3 = 3√3 Surds can only be added or subtracted as multiples of the same root: it is not √63.
Rationalise the denominator of 12/√3Extended: Multiply top and bottom by √3: 12√3/3 = 4√3
Rationalise the denominator of 2/(√5 − 1)Extended: Multiply top and bottom by the conjugate √5 + 1: 2(√5 + 1)/(5 − 1) = (√5 + 1)/2
Simplify √50 and √8 × √2Extended: √50 = √(25 × 2) = 5√2 √8 × √2 = √16 = 4