Scale factors, scale drawings and maps

(a) $2$ km

1. $4 \times 50\,000 = 200\,000$ cm.
2. $200\,000$ cm $= 2000$ m $= 2$ km.

• M1 Working out $4 \times 50\,000 = 200\,000$ (cm).
• A1 The correct answer, $2$ km.

(b) $7$ cm

1. $3.5$ km $= 350\,000$ cm.
2. $350\,000 \div 50\,000 = 7$ cm.

• M1 Converting $3.5$ km to $350\,000$ cm.
• A1 The correct answer, $7$ cm.

(a) $5.2$ m

1. $$6.5\times 80$$
2. $$520/100$$
3. Multiply a plan length by 80: 6.5 $\times$ 80 = 520 cm.
4. Divide by 100 to convert centimetres to metres: 5.2 m.

• P1 Establishing $6.5\times 80$ or an equivalent valid method.
• P1 Establishing $520/100$ or an equivalent valid method.
• A1 Correct answer: $5.2$ m

(a) $1.8$ km

1. $$7.2\times 25000$$
2. $$180000/100000$$
3. Actual distance = 7.2 $\times$ 25 000 = 180 000 cm.
4. There are 100 000 cm in 1 km, so the distance is 1.8 km.

• P1 Establishing $7.2\times 25000$ or an equivalent valid method.
• P1 Establishing $180000/100000$ or an equivalent valid method.
• A1 Correct answer: $1.8$ km

(a) $0.84$ km

1. Scale the map length to real centimetres.
   $$21/5\times 20000$$
2. Convert centimetres to kilometres.
   $$84000/100000$$
3. Therefore $0.84$ km.

• P1 Scale the map length to real centimetres.
• P1 Convert centimetres to kilometres.
• A1 Correct answer: $0.84$ km

(b) $1.68$ km

1. Include both outward and return paths.
   $$2\times (21/25)$$
2. Therefore $1.68$ km.

• P1 Include both outward and return paths.
• A1 Correct answer: $1.68$ km

(a) $1:50$

1. Convert the real length to centimetres and divide by its drawn length.
   $$500/10$$
2. Therefore $1:50$.

• M1 Convert the real length to centimetres and divide by its drawn length.
• A1 Correct answer: $1:50$

(b) $3.75$ m

1. Multiply by the scale and convert to metres.
   $$7.5\times 50/100$$
2. Therefore $3.75$ m.

• M1 Multiply by the scale and convert to metres.
• A1 Correct answer: $3.75$ m

(a) $9$ cm

1. Find the total route length at the given scale.
   $$7\times 100000/50000$$
2. Subtract the part already drawn.
   $$14-5$$
3. Therefore $9$ cm.

• P1 Find the total route length at the given scale.
• P1 Subtract the part already drawn.
• A1 Correct answer: $9$ cm

(b) Longer: each map centimetre represents half the real distance, so the route takes twice as many centimetres.

1. Longer: each map centimetre represents half the real distance, so the route takes twice as many centimetres.

• C1 Correct conclusion with supporting reasoning: Longer: each map centimetre represents half the real distance, so the route takes twice as many centimetres.

(a) $12$ m

1. $6 \times 200 = 1200$ cm $= 12$ m.

• B1 The correct answer, $12$ m.

(b) £486

1. Real width: $4.5 \times 200 = 900$ cm $= 9$ m.
2. Area: $12 \times 9 = 108\ \text{m}^2$.
3. Cost: $108 \times 4.50 = £486$.

• P1 Finding the real width, 9 m.
• P1 Finding the real area, $108\ \text{m}^2$.
• P1 Multiplying their area by 4.50.
• A1 The correct answer, £486.

(a) $5$ cm

1. Real length: $8 \times 25\,000 = 200\,000$ cm.
2. On map B: $200\,000 \div 40\,000 = 5$ cm.

• P1 Finding the real length, $200\,000$ cm (or 2 km).
• A1 The correct answer, $5$ cm.

(b) $0.75\ \text{km}^2$

1. On map A, $1$ cm represents $25\,000$ cm $= 0.25$ km.
2. So $1\ \text{cm}^2$ represents a square $0.25$ km by $0.25$ km.
   $$1 \text{ cm}^2 \to 0.25^2 = 0.0625 \text{ km}^2$$
3. $$12 \times 0.0625 = 0.75 \text{ km}^2$$

• P1 Writing 1 cm represents $0.25$ km (or 250 m).
• P1 Squaring the length scale: $1\ \text{cm}^2$ represents $0.0625\ \text{km}^2$ (or $62\,500\ \text{m}^2$).
• A1 $0.75\ \text{km}^2$.