B2. Operations

Example

Evaluate 12 + 3 × (8 − 5) ÷ 9, then solve: 4 shirts at $18 with 25% off the total — what do you pay?

1. Brackets: 8 − 5 = 3. Then 3 × 3 = 9, 9 ÷ 9 = 1. Then 12 + 1 = 13.
2. Story order: 4 × 18 = 72; 25% of 72 = 18; 72 − 18 = $54.
3. Check units at each step: shirts → dollars → dollars.

Example

Without calculating, name 60% as a fraction and $\frac{7}{10}$ as a percent.

1. 60% = 3 × 20% = $\frac{3}{5}$.
2. $\frac{7}{10}$ = 7 × 10% = 70%.
3. Build from the ones you know: tenths and fifths cover most cases.

Example

Increase 200 by 10%, decrease it by 25%, and increase it by 100%.

1. 10% of 200 = 20. Increase: 200 + 20 = 220.
2. 25% of 200 = 50. Decrease: 200 − 50 = 150.
3. 100% of 200 = 200. Increase: 400 — doubling. Also handy: 1% = 2, 5% = 10, 50% = 100.

Example

Compute (-3) + 2 and (-3) − 2 on the line.

1. Start at -3. Add 2: move 2 right → -1.
2. Start at -3. Subtract 2: move 2 left → -5.
3. Equation form: -3 + 2 = -1; -3 − 2 = -5.

Example

Add $\frac{2}{3} + \frac{1}{4}$ and subtract $\frac{5}{6} - \frac{3}{8}$.

1. LCM of 3 and 4 is 12: $\frac{8}{12} + \frac{3}{12} = \frac{11}{12}$.
2. LCM of 6 and 8 is 24: $\frac{20}{24} - \frac{9}{24} = \frac{11}{24}$.
3. Use the lowest common denominator; simplify the result if you can.

Example

Find the GCF of 24 and 36, and the LCM of 4, 6 and 10.

1. Prime factors: 24 = 2×2×2×3; 36 = 2×2×3×3. Shared: 2×2×3 = 12 → GCF 12.
2. LCM: take each prime the most times it appears in any number: 2×2 (from 4), 3 (from 6), 5 (from 10) = 60.

Example

Write 3 × 3 × 3 in exponential form and evaluate it. Then expand $10^{4}$.

1. $3^{3}$ — base 3, exponent 3. Value: 27.
2. $10^{4}$ = 10 × 10 × 10 × 10 = 10 000.
3. The exponent counts the factors, not the multiplications: $3^{3}$ is not 3 × 3.

Example

Find $\frac{1}{2} \times \frac{2}{3}$ and $\frac{1}{2} \div \frac{1}{4}$.

1. Multiply tops and bottoms: $\frac{2}{6} = \frac{1}{3}$. Picture: half of two thirds.
2. Divide: how many quarters in a half? 2. Or multiply by the reciprocal: $\frac{1}{2} \times \frac{4}{1} = 2$.
3. Multiplying by a proper fraction shrinks; dividing by one grows.

Example

Compute 1.2 × 0.4 and 1.2 ÷ 0.5.

1. Multiply as whole numbers, then place the point: the product has as many decimal places as the factors combined → 0.48.
2. ÷ 0.5: scale both by 10 → (12) ÷ 5 = 2.4.
3. Estimate first: 1.2 × 0.4 is a bit less than half of 1.2.

Example

Is the pencil table proportional? Is a taxi at $4 plus $2 per km proportional? Then find the cost of 8 pencils.

1. Pencils: cost ÷ count is always 3 → proportional (constant ratio, starts at 0).
2. Taxi: 1 km costs 6, 2 km cost 8 — ratio changes → not proportional (the $4 start).
3. 8 pencils: 8 × 3 = $24.

Summary

• Brackets, then × ÷, then + −; story order in problems.
• Know the equivalents table.
• Mental percents from 10%, 25%, 50%.
• Integers: right for +, left for −; subtracting a negative adds.
• LCD to add fractions; GCF and LCM from primes.
• Exponents count factors.
• Proportional = constant ratio through zero.