B2. Operations

Example

Tickets cost $12.50 each. A group of 7 buys tickets and pays with $150. How much is left?

1. Cost: 7 × 12.50. Split: 7 × 12 = 84; 7 × 0.50 = 3.5. Total 84 + 3.5.
2. Left: 150 − (that total).
3. Check by adding the cost back to what is left: it should make 150.

Example

Find 7 × 11 two ways, and write its division facts.

1. Split: 7 × 11 = 7 × 10 + 7 × 1 = 70 + 7 = 77.
2. Or double: if you know 7 × 11 halves, double it.
3. Division: 77 ÷ 11 = 7; 77 ÷ 7 = 11.

Example

Compute 82 × 0.1 and 82 × 0.01. Then estimate 4.87 + 2.19.

1. 82 × 0.1 = 8.2. 82 × 0.01 = 0.82.
2. Estimate: 4.87 ≈ 5, 2.19 ≈ 2, so about 7.
3. Exact is 7.06 — the estimate tells you the answer is near 7, not 70 or 0.7.

Example

A town's population was 48120 and grew by 35275. What is it now? A jug held 3.75 L and 1.8 L was poured out. How much is left?

1. Whole numbers: 48120 + 35275 = 83395, columns with regrouping.
2. Decimals: line up the points; 1.8 is 1.80. 3.75 − 1.80 = 1.95 L.
3. Check: 1.95 + 1.80 = 3.75. ✓

Example

Add $\frac{1}{5} + \frac{1}{5}$, then subtract $\frac{2}{5} - \frac{1}{5}$.

1. Same denominator → same piece size. Add the numerators: $\frac{2}{5}$.
2. Subtract the numerators: $\frac{1}{5}$.
3. The denominator never changes — you are counting pieces of one size.

Example

Multiply 46 × 15 with the area model, then connect it to the algorithm.

1. Split: 46 = 20 + 26; 15 = 10 + 5.
2. Four parts: 20 × 10 = 200; 20 × 5 = 100; 26 × 10 = 260; 26 × 5 = 130.
3. Add: 690. The written algorithm computes the same partial products in two rows.

Example

Divide 432 ÷ 12, then 500 ÷ 12, expressing the remainder.

1. 432 ÷ 12: 12 × 30 = 360, leaves 72; 12 × 6 = 72. Quotient 36.
2. 500 ÷ 12: 12 × 41 = 492, remainder 8.
3. Express it: 41 R 8, or 41 $\frac{8}{12}$ = 41 $\frac{2}{3}$, or ≈ 41.67 — choose what the situation needs (buses need a whole number, rounded up).

Example

Find **3 × $\frac{1}{4}$ and 3 ÷ $\frac{1}{4}$**.

1. 3 × $\frac{1}{4}$ = $\frac{3}{4}$: 3 pieces of one 4th.
2. 3 ÷ $\frac{1}{4}$ asks: how many 4ths are in 3 wholes? 4 per whole → 12.
3. Multiplying by a unit fraction makes a smaller number; dividing by one makes a bigger number.

Example

Paint mixes 2 red : 3 blue. Write two equivalent ratios. If 6 red are used, how many blue?

1. Multiply both parts: 4 : 6, 6 : 9.
2. 6 red is 3 × 2, so blue is 3 × 3 = 9.
3. A rate is a ratio with units: $3 per kg → $9 for 3 kg, the same scaling.

Summary

• Split, then check with the inverse.
• Facts to 12 × 12.
• × 0.1 shifts right one place; estimate decimals by rounding.
• Line up decimal points; like fractions add numerators.
• Area model = the algorithm's partial products.
• Ratios and rates scale both parts.