Rational Number Arithmetic

6.C.1Divide fractions01/04

Divide fractions by fractions.

Step01of 04

"Keep, change, flip." Keep the first fraction. Change division to multiplication. Flip the second (use its reciprocal). Then multiply normally.

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a d}{b c}

6.C.1 — Divide fractions by fractions.

Step 01 of 04

"Keep, change, flip." Keep the first fraction. Change division to multiplication. Flip the second (use its reciprocal). Then multiply normally.

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a d}{b c}

Step 02 of 04

Worked example. $\frac{2}{3} \div \frac{4}{5}$ .

= \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}

Step 03 of 04

Why? Dividing by 4/5 = "how many 4/5's fit?" If 4/5 fits 5/4 times into 1, it fits $(2/3) \times (5/4)$ times into 2/3. Multiplying by the reciprocal IS the operation.

Step 04 of 04

Mixed number example. $2 \frac{1}{2} \div 1 \frac{1}{4}$ .

Convert to improper: $5/2 \div 5/4 = 5/2 \times 4/5 = 20/10 = 2$ .

Key insight

"Keep, change, flip." Mixed numbers convert to improper first. The "÷ = × reciprocal" rule covers all fraction division — unit fractions or otherwise.

6.C.2 — Multi-digit division (standard algorithm).

Step 01 of 04

Long division with multi-digit divisors. Same divide-multiply-subtract-bring down rhythm; the trick is ESTIMATING quotient digits with bigger divisors.

Step 02 of 04

Worked example. 4,752 ÷ 18.

18 into 4? No. Into 47? About 2 (18 × 2 = 36; 18 × 3 = 54 too big). Write 2. 47 − 36 = 11. Bring down 5 → 115. 18 into 115? About 6 (18 × 6 = 108). Write 6. 115 − 108 = 7. Bring down 2 → 72. 18 into 72? Exactly 4. Write 4. 72 − 72 = 0.

4, 752 \div 18 = 264

Step 03 of 04

Estimating with multi-digit divisors. Round the divisor to the nearest 10. 18 ≈ 20. Then estimate: how many 20s fit in 47? About 2. Try 2.

Step 04 of 04

Adjust if your guess is off. Test by multiplying: if your guess × divisor > the dividend chunk, try one less. If there's still room after subtracting, try one more.

Key insight

Long division procedure stays the same. Estimate quotient digits by rounding the divisor. Verify by multiplying back. The procedure is mechanical once you trust it.

6.C.3 — Decimal arithmetic — all four operations.

Step 01 of 04

Add / subtract decimals. Line up decimal points, pad with trailing zeros if needed, operate, decimal in answer goes straight down.

5.4 + 2.36 = 5.40 + 2.36 = 7.76

Step 02 of 04

Multiply decimals. Multiply as whole numbers, then count total decimal places in factors and place the decimal in the answer.

0.4 \times 0.05 = 0.020 = 0.02 (1 + 2 = 3 decimal places)

Step 03 of 04

Divide decimals. Shift both numbers' decimals so the divisor is a whole number, then divide normally.

6.4 \div 0.4 = 64 \div 4 = 16

Step 04 of 04

Mixed example. Find $(1.2 \times 0.5) + 0.7$ .

1.2 \times 0.5 = 0.60

0.60 + 0.70 = 1.30

Key insight

All four decimal operations build on whole-number arithmetic. Track decimal places carefully (line up for +/−; count for ×; shift for ÷). Always check the result feels reasonable.

6.C.4 — GCF and LCM.

Step 01 of 04

Greatest Common Factor (GCF) = the LARGEST number that divides EVENLY into all given numbers.

Least Common Multiple (LCM) = the SMALLEST number that is a multiple of all given numbers.

Step 02 of 04

List method. List factors of each, find the biggest shared one.

Factors of 12	→1, 2, 3, 4, 6, 12
Factors of 18	→1, 2, 3, 6, 9, 18
GCF	→6 (largest in both lists)

Step 03 of 04

For LCM, list multiples and find the smallest shared one.

Multiples of 4	→4, 8, 12, 16, 20, 24, ...
Multiples of 6	→6, 12, 18, 24, 30, ...
LCM	→12 (smallest in both lists)

Step 04 of 04

Why these matter.

Reducing fractions	→divide top and bottom by GCF
Common denominator	→use LCM
Real life	→sets of items in different group sizes (LCM)

Key insight

GCF (greatest COMMON FACTOR) divides both. LCM (least COMMON MULTIPLE) is divisible by both. Use GCF for simplifying fractions; LCM for common denominators.

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