Direct and inverse proportion: tables and relationships

(a) £75

1. One ticket costs $45 \div 6 = £7.50$.
2. 10 tickets cost $10 \times 7.50 = £75$.

• M1 Finding the cost of one ticket, £7.50 (or of 2 tickets, £15).
• A1 The correct answer, £75.

(b) $750$ g

1. For 2 people: $150$ g. For 10 people: $5 \times 150 = 750$ g.

• M1 Scaling correctly, such as $300 \div 4 = 75$ g per person or $\times 2.5$.
• A1 The correct answer, $750$ g.

(a) $630$ g

1. $$420/6$$
2. $$70\times 9$$
3. The ratio of oats to portions must stay the same.
4. 420 $\div$ 6 = 70 g per portion; 9 $\times$ 70 = 630 g.

• P1 Establishing $420/6$ or an equivalent valid method.
• P1 Establishing $70\times 9$ or an equivalent valid method.
• A1 Correct answer: $630$ g

(a) $35$ minutes

1. $$84/12$$
2. $$245/7$$
3. The machine makes 84 $\div$ 12 = 7 parts per minute.
4. Time = 245 $\div$ 7 = 35 minutes.

• P1 Establishing $84/12$ or an equivalent valid method.
• P1 Establishing $245/7$ or an equivalent valid method.
• A1 Correct answer: $35$ minutes

(a) $10.5$ hours

1. $$6\times 14$$
2. $$84/8$$
3. The job requires 6 $\times$ 14 = 84 pump-hours.
4. With 8 pumps, time = 84 $\div$ 8 = 10.5 hours.

• P1 Establishing $6\times 14$ or an equivalent valid method.
• P1 Establishing $84/8$ or an equivalent valid method.
• A1 Correct answer: $10.5$ hours

(a) $4$

1. $$72/6$$
2. $$72/9$$
3. Use the inverse-proportion equation at both values.
4. At x = 6, y = 12; at x = 9, y = 8.
5. The decrease is 12 $-$ 8 = 4.

• P1 Establishing $72/6$ or an equivalent valid method.
• P1 Establishing $72/9$ or an equivalent valid method.
• A1 Correct answer: $4$

(a) $3$

1. $$150\times 60/8$$
2. $$1125/500$$
3. Scale the flour by 60/8: 150 $\times$ 60/8 = 1125 g.
4. 1125 $\div$ 500 = 2.25 bags; whole bags must be bought.
5. Round up to 3 bags.

• P1 Establishing $150\times 60/8$ or an equivalent valid method.
• P1 Establishing $1125/500$ or an equivalent valid method.
• A1 Correct answer: $3$

(a) $20$ hours

1. Find the output of one machine in one hour.
   $$9000 \div 6 \div 10 = 150 \text{ bottles}$$
2. Four machines make:
   $$4 \times 150 = 600 \text{ bottles per hour}$$
3. $$12\,000 \div 600 = 20 \text{ hours}$$

• P1 Finding the rate for one machine, 150 bottles per hour (or 60 machine-hours for 9000 bottles).
• P1 Finding the rate for 4 machines (600 per hour), or the machine-hours for 12 000 bottles (80).
• A1 The correct answer, $20$ hours.

(a) $6$

1. $$12\times (15-6)$$
2. $$108/6$$
3. $$18-12$$
4. The job requires 12 $\times$ 15 = 180 worker-days; 72 have been completed.
5. There are 108 worker-days left and only 6 days available, so 18 workers are needed.
6. Additional workers = 18 $-$ 12 = 6.

• P1 Establishing $12\times (15-6)$ or an equivalent valid method.
• P1 Establishing $108/6$ or an equivalent valid method.
• P1 Establishing $18-12$ or an equivalent valid method.
• A1 Correct answer: $6$