B2. Operations

Example

A recipe for 4 uses 1.5 cups of flour. You cook for 10 and have 5 cups. Enough?

1. Rate: 1.5 ÷ 4 = 0.375 cups per person.
2. For 10: 0.375 × 10 = 3.75 cups.
3. 3.75 < 5. Yes, with 1.25 to spare. Check each step's units.

Example

Test 540 against every rule.

1. Even → 2. Ends in 0 → 5, 10. Digit sum 9 → 3 and 9. 2 and 3 → 6.
2. Last two digits 40 ÷ 4 = 10 → 4. Last three 540 ÷ 8 = 67.5 → not 8.
3. 540 is divisible by 2, 3, 4, 5, 6, 9, 10 — not 8.

Example

Find 10%, 5%, 1%, 15%, 25% and 50% of 400.

1. 10%: 40. 5%: 20. 1%: 4.
2. 15%: 40 + 20 = 60.
3. 25%: 100. 50%: 200.

Example

Add 78905 + 156 789. Then subtract 7.25 − 3.875.

1. Estimate first: about 78905 + 157 000.
2. Columns with regrouping: 235694.
3. Decimals: 7.250 − 3.875 = 3.375 — pad with zeros, line up the points.

Example

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

1. Common denominator 6: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$.
2. Common denominator 12: $\frac{9}{12} - \frac{2}{12} = \frac{7}{12}$.
3. Change the pieces to one size first; then add or subtract the counts.

Example

Write 60 as a product of primes with a factor tree.

1. 60 = 6 × 10. Split again: 6 = 2 × 3; 10 = 2 × 5.
2. Leaves are all prime: 60 = 2 × 2 × 3 × 5.
3. Check by multiplying back: 2 × 2 × 3 × 5 = 60. ✓ Any starting split gives the same primes.

Example

Multiply 236 × 0.5 and 236 × 0.3.

1. × 0.5 is half: 118.
2. × 0.3: 236 × 3 = 708, then one decimal place: 70.8.
3. Count decimal places in the factors; the product has the same number.

Example

Divide 240 ÷ 0.4 and 7 ÷ 0.5.

1. Ask: how many 0.4s in 240? Scale both by 10: 2400 ÷ 4 = 600.
2. 7 ÷ 0.5: how many halves in 7? 14.
3. Dividing by a number less than 1 gives a bigger answer.

Example

Find **6 × $\frac{2}{3}$**.

1. Two thirds of 6: one third is 2, so two thirds is 4.
2. Or multiply then divide: 6 × 2 = 12; ÷ 3 = 4.
3. The product is smaller than 6 — a proper fraction shrinks.

Example

Find **6 ÷ $\frac{3}{4}$**.

1. Ask: how many three-quarters fit in 6?
2. 6 has 24 quarters; groups of 3 quarters: 24 ÷ 3 = 8.
3. Dividing by a proper fraction gives a bigger answer.

Example

Divide 3.6 ÷ 4 and 0.375 ÷ 3.

1. 3.6 ÷ 4 = 0.9: divide as whole numbers, keep the decimal point in place.
2. 0.375 ÷ 3 = 0.125.
3. Check by multiplying: 0.9 × 4 = 3.6. ✓

Example

Juice is mixed 2 parts concentrate : 3 parts water. How much water for 8 cups of concentrate? A $400 bike is 25% off — what is the discount?

1. 8 is 4 × 2, so water is 4 × 3 = 12 cups.
2. 25% of 400 = 100. Discount $100.
3. Ratios scale both parts; percents are a ratio out of 100.

Summary

• Divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10.
• Percents from 10%.
• Common denominators before adding.
• Factor trees multiply back.
• Decimal places count; dividing by less than 1 grows.
• Ratios scale both parts.