Arithmetic
Operations, fractions, percents, ratios, powers, number properties and word problems. No calculator, so simplifying and estimating are your key skills.
Operations, fractions, percents, ratios, powers, number properties and word problems. No calculator, so simplifying and estimating are your key skills.
Operations, fractions and decimals
Order of operations
Brackets → powers → × and ÷ → + and −, left to right.
Adding fractions
Make the bottoms equal, add the tops, simplify.
Multiplying and dividing fractions
Multiply tops and bottoms. To divide: flip the second and multiply.
Compound fractions and mixed numbers
Compound fraction: simplify the top and the bottom separately, then divide. Mixed number: turn it into an improper fraction before multiplying or dividing. Decimal: turn it into a fraction (0.75 = ¾).
A fraction of a number
Divide by the bottom, then multiply by the top.
Multiplying decimals
Multiply as whole numbers, then place the point using the total decimal places.
Ordering numbers in different forms
Turn every choice into a decimal: a fraction by dividing, a percent by dividing by 100, a root by working it out, then compare.
Estimation
Round each number to an easy one, then work it out; the choices are usually far apart.
Percentages
A percent of a number
Turn the percent into a fraction: a% = a 100
Percent change (rise, fall, profit, loss)
difference original × 100, always divide by the original.
Up then down
Apply the two changes one after the other; the second uses the new price.
The price before a discount
New price = (100 − discount)% of the original, so divide, don’t add.
Price after a discount
Discount = rate × price, then subtract. Or directly: price × (1 − rate).
Ratio and proportion
Sharing in a ratio
Add the ratio parts and divide the total to find one part.
Three-part ratio (triangle angles)
Triangle angles total 180; divide it by the sum of parts.
Direct proportion
One goes up, the other goes up: find one unit, then multiply.
Inverse proportion
More workers → fewer days: the product stays the same.
Powers and roots
Power rules
a m × a n = a m + n, a m ÷ a n = a m − n
Zero and negative powers
Any non-zero number to the power 0 is 1. A negative power means flip it.
Simplifying roots
Split the number into a perfect square × the rest.
Multiplying and dividing roots
a × b = a × b, a ÷ b = a b. Simplify each root to number × root so the common root shows.
Roots of fractions and mixed numbers
Turn a mixed number into an improper fraction first, then root the top and the bottom.
Cube roots
Learn the cubes 1 to 10 (memory tables chapter). The cube root of a negative is negative, and the root of a product = the product of the roots.
A root in the denominator
Multiply the top and bottom by the same root to clear it: √a × √a = a.
Roots of fractions, decimals and cubes
a b = a b, 0.09 = 0.3
Number properties
Odd and even
even ± even = even, odd ± odd = even, even ± odd = odd, × even = even, odd × odd = odd.
Remainders
number = divisor × quotient + remainder, and the remainder is smaller than the divisor.
Least common multiple
Phrases like “ together again ” or “at the same time again” mean LCM.
Greatest common divisor
“ Largest number that divides ” or “largest equal groups” mean GCD.
Prime numbers
A prime has exactly two divisors: 1 and itself. 1 is not prime; 2 is the only even prime.
Sequences and time
The next term
Work out the differences; if they aren’t constant try: ratio, differences of differences, squares, alternating, sum of the previous two.
The nth term
nth term = first + (n − 1) × difference
Clock angle
|30 × hours − 5.5 × minutes|; if above 180, subtract from 360.
Days of the week
Divide the number of days by 7 and move on by the remainder.
Word problems
Speed, distance, time
distance = speed × time
Average speed for a round trip
2 × v1 × v2 v1 + v2, not the average of the speeds.
Meeting head-on
Moving toward each other: add the speeds (closing speed), then divide the distance.
Working together
time together = a × b a + b (a and b are the solo times).
Filling and draining a tank
time = a × b b − a (a to fill, b to drain).
Ages
Call the unknown age x, and add the years to both people for the future.
The mean after adding a number
Turn both means into totals and subtract.
Unit conversion
Learn the conversion table in “Memory tables”. Careful: 1 m² = 10,000 cm², not 100.
Production rates
Multiply the rate by the number of periods: a week is 7 days, a year 52 weeks or 12 months, a day 24 hours.
- In discounts and rises the percent is taken from the original.
- Read exactly what is asked: “left” or “used”? “larger” or “smaller”?
Ideas straight from the exam banks
Sums of runs and counting numbers
1 + 2 + … + n = n(n + 1) 2, odd numbers 1 + 3 + … = n² (n = number of terms). From a to b inclusive = b − a + 1; strictly between = b − a − 1.
Matches, gifts, trees and queues
League matches = n(n − 1) ÷ 2 per round. Gifts = n(n − 1). Cuts = pieces − 1. Trees along a road = length ÷ gap + 1, around a circle without the +1. Queue = both positions − 1 (in a circle − 2). Joined tables = 2n + 2.
Percentages in several steps
Two changes a% then b%: total = a + b + (a × b) ÷ 100 (a cut is negative). “Increased by 4 times itself” = 5×, “made 4 times as large” = 4×.
Divisibility and last digits
3 and 9: digit sum. 4: last two digits. 8: last three. 6 = 2 and 3. 11: alternating digit sum. Units digits of powers repeat every 4: divide the power by 4 and use the remainder.
Catching up and km/h to m/s
Catch-up time = first speed × head start ÷ speed difference. Same direction: subtract the speeds; opposite: add them. km/h to m/s: × ⁵⁄₁₈.
Three-way proportion, zakat and equal salaries
Find what one worker does in one day, then multiply. Zakat = money ÷ 40. Salaries meet after salary gap ÷ raise gap years.
Powers: different bases and common factors
3 x + 2 = 5 x + 2 → the power is 0 → x = −2. Repeated terms: 3 4 + 3 4 + 3 4 = 3 5. Sums: take out the common factor.
“All but” riddles and a changing mean
Add the “all but” numbers to get twice the total. Adding marks to one student raises the mean by extra ÷ count.
Remainder of the remainder, density, boxes, pegs, round tables and map scale
Remainder of the remainder: multiply what is left each time (half, then a third of the rest → 1 2 × 2 3 = 1 3), then divide what remains by it. Density = people ÷ area. Boxes = a + ab + abc. Shared pegs = pieces + 1. Round table: (n − 1)!; a row: n!. Map: distance × scale ÷ 100000 = km.
Pay plus a prize, a ratio after newcomers, page numbers, multiples and gifts
Pay + a phone: subtract the two cases and the phone cancels: money difference ÷ hours difference = hourly pay. Ratio after newcomers: call the parts x and write the new ratio as an equation. Pages: up to 99 pages digits − 9 2 + 9, beyond that digits − 189 3 + 99. Multiples of d from a to b = ⌊b÷d⌋ − ⌊(a−1)÷d⌋. Gifts = n(n − 1), not halved.
Decimals and powers of ten
Multiplying: multiply without points, then add up the decimal places (0.3 × 0.03 = 0.009). Dividing: match the places, then drop the point (0.12 ÷ 0.04 = 12 ÷ 4 = 3). A fraction of one percent: 3 4 % = 0.0075. A square root halves the places (√0.0016 = 0.04), a cube root divides them by 3 (∛0.125 = 0.5).