B2. Operations

Example

Evaluate −6 + 4 × (−3) − 8 ÷ 2, then **$\frac{1}{2}$ of 40% of 150**.

1. × and ÷ first: 4 × (−3) = −12; 8 ÷ 2 = 4. Then −6 + (−12) − 4 = −22.
2. 40% of 150 = 60; half of 60 = 30.
3. Same rules for every number type; keep signs with their numbers.

Example

Write, from memory, the squares from 1 to 15 and their roots.

1. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
2. $\sqrt{169} = 13$, $\sqrt{196} = 14$, $\sqrt{225} = 15$.
3. Knowing these makes root estimates and the Pythagorean theorem fast.

Example

Compute 3.45 × 100, 3.45 ÷ 10 and 3.45 × 1000 in your head.

1. × 100: shift the point two places right → 345.
2. ÷ 10: one place left → 0.345.
3. × 1000: three places right → 3450. Digits never change; only the point moves.

Example

Compute (-3) + 9, (-3) − 9 and 9 − (-3).

1. (-3) + 9 = 6: opposite signs — subtract sizes, keep the larger's sign.
2. (-3) − 9 = -12: subtracting a positive moves left.
3. 9 − (-3) = 12: subtracting a negative adds its opposite.

Example

Compute $2\frac{1}{3} + 1\frac{3}{4}$ and $\frac{5}{6} - \frac{7}{8}$.

1. Mixed: wholes 2 + 1 = 3; fractions $\frac{4}{12} + \frac{9}{12} = \frac{13}{12} = 1\frac{1}{12}$; total $4\frac{1}{12}$.
2. LCD 24: $\frac{20}{24} - \frac{21}{24} = -\frac{1}{24}$. A negative fraction is a fine answer.
3. Always the lowest common denominator; simplify at the end.

Example

Compute $1\frac{1}{2} \times 4$, $\frac{2}{3} \times \frac{3}{5}$ and $2\frac{1}{2} \div \frac{1}{2}$.

1. Mixed → improper: $\frac{3}{2} \times 4 = \frac{12}{2} = 6$.
2. $\frac{2}{3} \times \frac{3}{5} = \frac{6}{15} = \frac{2}{5}$ — cancel a 3 first if you like.
3. $\frac{5}{2} \div \frac{1}{2} = \frac{5}{2} \times 2 = 5$: dividing by a half is doubling.

Example

Compute (7) × 5, then divide the result by 5, then by (−5).

1. (7) × 5 = 35.
2. 35 ÷ 5 = 7 — the inverse returns the start.
3. 35 ÷ (−5) = -7: dividing by the opposite flips the sign.

Example

Compare $3 per item with $4 plus $2 per item. Which is proportional?

1. 3 per item: cost ÷ items is always 3, and 0 items cost 0 → proportional.
2. $4 + $2 per item: 1 item costs 6, 2 cost 8 — the ratio changes → not proportional.

Example

Then find the cost of 12 items in the proportional one.

1. 12 items: 12 × 3 = $36. Or scale: 4 items cost 12, so 12 cost 3 × 12.

Summary

• Order of operations holds for every number type.
• Squares to 15².
• Powers of ten move the point.
• Integers: sign rules for + − × ÷.
• Fractions: LCD to add; improper form to multiply and divide.
• Proportional = constant ratio through zero.