Comparing numbers: difference or ratio
There are two honest ways to compare 15 with 12.
- Difference: 15 − 12 = 3. It says how many more, in the same unit.
- Ratio: 15 ÷ 12 = 1.25. It says how many times as big, with no unit. 1.25 means +25%.
Use the difference when the unit matters (3 more students). Use the ratio when you want to compare things of different size: a rise of ₹10 is big for a ₹40 pen and tiny for a ₹4000 bag.
Fractions, powers and orders of magnitude
Powers of ten shorten big and small numbers. 10³ = 1000 (thousand), 10⁶ = 1 000 000 (million), 10⁰ = 1, 10⁻² = 1/100 = 0.01. Two rules: 10ᵃ × 10ᵇ = 10ᵃ⁺ᵇ and 10ᵃ ÷ 10ᵇ = 10ᵃ⁻ᵇ.
The order of magnitude of a number is the nearest power of ten. A cricket ground is about 10² m across; the distance to the Moon is about 4 × 10⁸ m, so it is roughly 10⁶ times bigger. When you add fractions, find a common denominator first (1/4 + 1/6 = 3/12 + 2/12 = 5/12). To divide by a fraction, multiply by its flip.
Unit conversions
Write the unit change as a factor of 1 and multiply. 1 km = 1000 m, 1 h = 3600 s, 1 L = 1000 cm³, 1 m² = 10 000 cm² (the factor is squared for area) and 1 m³ = 10⁶ cm³ (cubed for volume).
Speed: 36 km/h = 36 × 1000 m ÷ 3600 s = 10 m/s. Shortcut: divide km/h by 3.6 to get m/s, and multiply m/s by 3.6 to get km/h.
Percent changes: successive and reciprocal
A change of p% turns a value into value × (1 + p/100). The number 1 + p/100 is the factor. Rise of 20%: × 1.2. Fall of 20%: × 0.8.
Successive changes multiply their factors. +20% then −20%: 1.2 × 0.8 = 0.96, so the result is 4% lower. The second percent is taken of the new, bigger total, which is why the changes do not cancel.
Reciprocal change (the change that undoes another): the factor is the flip. After +25% (× 1.25) the undo factor is 1/1.25 = 0.8, a fall of 20%.
Geometric growth means the same factor again and again: 3, 6, 12, 24 has factor 2. Compare arithmetic growth, which adds the same amount: 3, 6, 9, 12. If a quantity grows 5% every year, after n years it is × 1.05ⁿ.
Expanding, factorising and special products
Expanding opens brackets: a(b + c) = ab + ac. Factorising does the reverse: 6x + 9 = 3(2x + 3).
Three special products are worth knowing by heart: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b² and (a + b)(a − b) = a² − b². The 3D shows why there is a middle part 2ab: the big square has two equal rectangles of area ab.
They give fast arithmetic: 99² = (100 − 1)² = 9801 and 52 × 48 = 50² − 2² = 2496.
Making one letter the subject of a formula
Do the same thing to both sides until the letter you want stands alone. For v = u + at, subtract u: v − u = at, then divide by a: t = (v − u)/a. For A = ½bh, multiply by 2 and divide by b: h = 2A/b. Work backwards in the order the formula was built.
Zero-product equations and signs of expressions
A product is zero only if one factor is zero. So (x − 3)(x + 2) = 0 gives x = 3 or x = −2.
To find the sign of a factorised expression, mark where each factor is zero on a number line and test one point in each part. For (x − 1)(x + 4): the zeros are −4 and 1. At x = 0 the value is −4 (negative), so the expression is negative between −4 and 1 and positive outside. A linear expression ax + b changes sign once, at x = −b/a.
Reading graphs, charts, tables and trees fast
For a line y = mx + c, c is where it meets the y-axis and m (the slope) is how much y rises for 1 step in x. A bar chart or pie chart is read by comparing sizes first and numbers second. For a conditional probability from a table, divide by the row or column you are told to restrict to: P(A given B) = (A and B) ÷ (all B). On a tree, multiply along the branches.
If a function's derivative is positive on an interval, the function goes up there; if negative, it goes down. So the sign of the derivative gives the table of variation.
Chained routine calculations
A chain is several steps one after another, like a price that falls 15% and then rises 10%. Write each step as one operation (× 0.85, then × 1.1), keep the intermediate result, and check at the end with an estimate: 1200 × 0.85 = 1020, then 1020 × 1.1 = 1122. Estimating first (about 1200 − 180 + 100) catches slips.
Key formulas and definitions
- New value = old value × (1 + p/100)
- Successive changes: factor = (1 + p₁/100) × (1 + p₂/100)
- Undo +p%: factor 1/(1 + p/100); +25% is undone by −20%
- Growth n times at p%: value × (1 + p/100)ⁿ
- (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b²
- (a + b)(a − b) = a² − b²
- ab = 0 ⇒ a = 0 or b = 0
- km/h ÷ 3.6 = m/s
Worked examples
1. Which rise is bigger in relative terms: 20 → 25 or 50 → 60?
Ratios: 25/20 = 1.25 (+25%) and 60/50 = 1.2 (+20%). The first rise is bigger in relative terms, although the second has the bigger difference (10 against 5).
2. The Moon is about 4 × 10⁸ m away and a cricket pitch is about 2 × 10¹ m long. How many times bigger is the distance?
Divide: (4 × 10⁸) ÷ (2 × 10¹) = 2 × 10⁷. So it is about 20 million times as far.
3. Convert 2.5 L to cm³, and 54 km/h to m/s.
1 L = 1000 cm³, so 2.5 L = 2500 cm³. 54 ÷ 3.6 = 15 m/s.
4. A price of 80 rises by 25%. What is the new price? By what percent must it fall to return to 80?
New price = 80 × 1.25 = 100. Undo factor = 1/1.25 = 0.8, so a fall of 20% (100 × 0.8 = 80).
5. A bank gives 10% interest in each of two years on 200. What is the total after two years?
Factor = 1.1 × 1.1 = 1.21. So 200 × 1.21 = 242. This is 21% more, not 20%.
6. Compute 99² and 52 × 48 without a calculator.
99² = (100 − 1)² = 10 000 − 200 + 1 = 9801. 52 × 48 = (50 + 2)(50 − 2) = 2500 − 4 = 2496.
7. Make h the subject of A = ½bh and find h when A = 30 and b = 12.
Multiply by 2: 2A = bh. Divide by b: h = 2A/b. So h = 60/12 = 5.
8. Solve (x − 4)(2x + 6) = 0.
A product is zero when a factor is zero. x − 4 = 0 gives x = 4. 2x + 6 = 0 gives x = −3. Solutions: 4 and −3.
9. Find where (x + 2)(x − 5) is positive.
The zeros are −2 and 5. At x = 0 the value is (2)(−5) = −10, negative between −2 and 5. So it is positive for x < −2 or x > 5.
10. A bicycle costs 1200. It falls 15% in a sale and then rises 10% a month later. Find the final price.
1200 × 0.85 = 1020. Then 1020 × 1.1 = 1122. Overall factor 0.935, so 6.5% lower.
11. Is 3, 6, 12, 24 geometric or arithmetic? What is the next term?
Ratios 6/3 = 12/6 = 24/12 = 2, so it is geometric with factor 2. The next term is 48. (Differences 3, 6, 12 are not equal, so it is not arithmetic.)
Common mistakes
- Thinking +20% then −20% gives the start again. The 20% fall is taken of the bigger number, so you end 4% lower.
- Adding successive percents (10% + 10% = 20%). You must multiply the factors: 1.1 × 1.1 = 1.21, so 21%.
- Undoing +25% with −25%. The undo percent is 20%, from the factor 1/1.25 = 0.8.
- Writing (a + b)² = a² + b². The middle part 2ab is missing; 3D blocks show two rectangles of ab.
- Solving (x − 3)(x + 2) = 6 by saying x − 3 = 6 or x + 2 = 6. The zero-product rule only works when the product equals 0.