Printable · GCSE Foundation · ages 14-16
Scale factors, scale diagrams and maps worksheet — GCSE Foundation
Fifteen questions on "scale factors, scale diagrams and maps" — DfE statement R2. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
Scale factors, scale diagrams and maps worksheet — GCSE Foundation
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- 1.A map has a scale of 1 : 100 000. Two railway stations are 8 km apart in real life. Work out the distance between the stations on the map, in centimetres.
- 2.A map has a scale of 1 : 40 000. A cycle path measures 7.5 cm on the map. Work out the real length of the cycle path, in kilometres.
- 3.A river 120 km long is drawn as a line 6 cm long on a map. Work out the scale of the map.
- 4.A map has a scale of 1 : 20 000. A different map of the same area has a scale of 1 : 80 000. A lake's shoreline is drawn 5 cm long on the first map. Work out the length of the same shoreline on the second map, in centimetres.
- 5.A rectangular room is drawn on a plan with a scale of 1 : 50. On the plan, the room measures 8 cm by 6 cm. Work out the real area of the room, in square metres.
- 6.A map has a scale of 1 : 1000. A road is 300 m long in real life. Work out the length of the road on the map, in centimetres.
- 7.A plan is drawn to a scale of 1 : 25. A real fence is 10 m long. Work out the length of the fence on the plan, in centimetres.
- 8.Map A has a scale of 1 : 25000 and Map B has a scale of 1 : 50000, both showing the same area. A real distance of 10 km is measured on each map. On which map does this distance appear as the longer length, and how long is it on that map, in centimetres?
- 9.A cartographer is choosing a scale for a map to show a small park in as much detail as possible. Which of these scales would show the MOST detail — 1 : 500, 1 : 5000, or 1 : 50 000? Give the correct scale and reason.
- 10.A scale drawing states '1 cm represents 25 m'. Which of these is the correct way to write this scale in the form 1 : n?
- 11.A scale model of a bridge is built at a scale of 1 : 120. The real bridge is 84 m long. Work out the length of the model, in centimetres.
- 12.Two scale plans of the same football pitch are drawn: Plan A has a scale of 1 : 1000 and Plan B has a scale of 1 : 200. A pitch length is 8 cm on Plan A. Work out the length of the same pitch on Plan B, in centimetres.
- 13.A map has a scale of 2 cm : 5 km. A footpath measures 8 cm on the map. Work out the real length of the footpath, in kilometres.
- 14.A scale drawing of a car park has a scale of 1 : 300. On the drawing, the car park has an area of 15 cm². Work out the real area of the car park, in square metres.
- 15.A plan is drawn to a scale of 1 : 500. Work out how many metres are represented by 1 cm on the plan.
Answer key
- (a) 8 cm — Method: a scale of 1 : n means the real distance is n times the distance on the map, so to go from the real distance back to the map distance, put both lengths in the same unit and then divide by the scale. Working: 1 km = 100 000 cm, so 8 km = 8 × 100 000 = 800 000 cm; 800 000 ÷ 100 000 = 8. Answer: 8 cm. The distractors: 800000 cm comes from converting the 8 km into centimetres and stopping there, so the division by the scale — the inverse operation the question asks for — is never done; 80 cm comes from taking a metre to be 1000 cm, which turns 8 km into 8 × 1000 × 1000 = 8 000 000 cm and gives 8 000 000 ÷ 100 000 = 80; 0.08 cm comes from taking a kilometre to be 1000 cm, which turns 8 km into 8000 cm and gives 8000 ÷ 100 000 = 0.08.
- (b) 3 km — Method: multiply by the scale factor to get the real length in centimetres, then convert to kilometres. Working: 7.5 × 40 000 = 300 000 cm. 300 000 ÷ 100 000 = 3 km. Wrong options: 30 km comes from dividing by 10 000 instead of 100 000 when converting to kilometres; 3000 km comes from dividing by 100 instead of 100 000; 0.3 km comes from dividing by 1 000 000, an extra factor of 10 too many.
- (a) 1 : 2 000 000 — Method: a scale is a ratio between two lengths written in the same unit, reduced so that the map distance is 1. Working: 120 km = 120 × 1000 × 100 = 12 000 000 cm, so the ratio is 6 : 12 000 000, and dividing both parts by 6 gives 1 : 2 000 000. Answer: 1 : 2 000 000. The distractors: 1 : 12 000 000 comes from writing the real length in centimetres without dividing by the 6 cm on the map; 1 : 200 000 comes from taking 120 km as 1 200 000 cm, one conversion step short, before dividing by 6; 1 : 20 000 comes from converting 120 km to 120 000 m and treating those metres as centimetres.
- (c) 1.25 cm — Method: find the real length using the first map's scale, then use the second map's scale to find its drawn length. Working: real length = 5 × 20 000 = 100 000 cm. On the second map: 100 000 ÷ 80 000 = 1.25 cm. Wrong options: 20 cm comes from inverting the ratio of the two scales (5 × 80 000 ÷ 20 000); 5 cm comes from wrongly assuming the length looks the same on both maps; 12.5 cm comes from dropping a zero from the second scale factor and dividing by 8 000 instead of 80 000 (100 000 ÷ 8 000).
- (a) 12 m² — Real length = 8 × 50 = 400 cm = 4 m. Real width = 6 × 50 = 300 cm = 3 m. Real area = 4 × 3 = 12 m². Scaling the plan area (8 × 6 = 48 cm²) by 50 instead of by 50 squared gives 48 × 50 = 2400 cm² = 0.24 m² — area scales by the square of the length scale factor, not the scale factor itself. Multiplying the real dimensions in centimetres, 400 × 300 = 120 000, and calling the result 120 000 m² mistakes square centimetres for square metres. Converting only the length to metres and leaving the width as 6 (treating centimetres as metres), 4 × 6 = 24, gives 24 m², from a scaling that was never finished.
- (d) 30 cm — Convert 300 m to centimetres: 300 m = 30 000 cm. The scale 1 : 1000 means the map length is the real length divided by 1000: 30 000 ÷ 1000 = 30, giving 30 cm. Dividing the unconverted 300 (metres, not centimetres) by 1000 gives 0.3 cm, far too small to be a real map measurement. Dividing by 10 000 instead of 1 000 gives 3 cm, ten times too small. Multiplying instead of dividing, 300 × 1000 = 300 000, gives 300 000 cm — using the scale the wrong way round, as if the map were bigger than real life.
- (d) 40 cm — Method: convert the real length to centimetres, then divide by the scale factor. Working: 10 m = 1000 cm. 1000 ÷ 25 = 40 cm. Wrong options: 0.4 cm comes from dividing 10 by 25 without converting metres to centimetres first; 25000 cm comes from multiplying instead of dividing (1000 × 25); 4 cm comes from converting 10 m to 100 cm (using ×10 instead of ×100) before dividing.
- (c) Map A, where the distance is 40 cm — 10 km = 1,000,000 cm. On Map A: 1000000 ÷ 25000 = 40 cm. On Map B: 1000000 ÷ 50000 = 20 cm. Since 40 cm is longer than 20 cm, the same real distance appears longer on Map A, the map with the smaller scale number. 'Map B, where the distance is 20 cm' has the correct working for Map B but names the wrong map as the one with the longer length. 'Map A, where the distance is 20 cm' correctly identifies Map A but pairs it with Map B's length. 'Map B, where the distance is 40 cm' correctly identifies Map A's length but attaches it to the wrong map.
- (b) 1 : 500 — smallest real distance per cm (5 m), most detail — Method: compare what one centimetre represents in real life for each scale — the scale with the smallest real distance per cm shows the most detail. Working: for 1 : 500, 1 cm represents 500 cm (5 m); for 1 : 5000, 1 cm represents 50 m; for 1 : 50 000, 1 cm represents 500 m. Since 5 m is the smallest, 1 : 500 shows the most detail. Wrong options: '1 : 50 000 — covers the largest real area' wrongly assumes covering more area means more detail, when it is the opposite; '1 : 5000 — the middle value' wrongly assumes the middle scale is automatically the most balanced; '1 : 500 — covers the largest real distance' picks the correct scale but states an incorrect fact, since 1 : 500 actually covers the smallest real distance per cm.
- (d) 1 : 2500 — Method: convert the real-world measurement to the same unit as the drawing (centimetres) before writing the ratio. Working: 25 m = 2500 cm, so the scale is 1 : 2500. Wrong options: 1 : 25 comes from not converting metres to centimetres at all; 1 : 250 comes from converting metres to centimetres using ×10 instead of ×100; 1 : 2.5 comes from converting in the wrong direction (treating 25 m as 2.5 cm).
- (c) 70 cm — Method: convert the real length to centimetres, then divide by the scale factor. Working: 84 m = 8400 cm. 8400 ÷ 120 = 70 cm. Wrong options: 0.7 cm comes from dividing 84 by 120 without converting metres to centimetres; 1,008,000 cm comes from multiplying instead of dividing (8400 × 120); 7 cm comes from converting 84 m to 840 cm (using ×10 instead of ×100) before dividing.
- (c) 40 cm — First find the real pitch length using Plan A's scale: 8 × 1000 = 8000 cm. Now find its length on Plan B using Plan B's scale: 8000 ÷ 200 = 40 cm. Assuming both plans use the same scale and leaving the length as 8 cm ignores that Plan B has a different scale. Using the ratio of the two scales upside down, 200 ÷ 1000 = 0.2, then 8 × 0.2 = 1.6, gives 1.6 cm — Plan B is drawn to a larger scale than Plan A, so lengths on it should be bigger, not smaller. Dividing by 2000 instead of 200 gives 4 cm, ten times too small.
- (b) 20 km — The scale 2 cm : 5 km means each 1 cm on the map represents 5 ÷ 2 = 2.5 km in real life. The footpath is 8 cm on the map, so its real length is 8 × 2.5 = 20 km. 40 km comes from multiplying 8 by 5 directly, ignoring that the scale's '2 cm' has to be divided out first: 8 × 5 = 40. 3.2 km comes from dividing 8 by 2.5 instead of multiplying: 8 ÷ 2.5 = 3.2. 5 km comes from multiplying 2.5 by the scale's '2' instead of by the footpath's 8 cm: 2.5 × 2 = 5.
- (d) 135 m² — Method: for area, the scale factor must be squared. Working: area scale = 300² = 90 000. 15 × 90 000 = 1,350,000 cm². Convert to m² by dividing by 10 000: 1,350,000 ÷ 10 000 = 135 m². Wrong options: 0.45 m² comes from using the linear scale factor (×300) instead of squaring it; 1,350,000 m² comes from forgetting to convert the answer from cm² to m²; 13,500 m² comes from dividing by 100 instead of 10 000 when converting units.
- (b) 5 m — A scale of 1 : 500 means 1 cm on the plan represents 500 cm in real life. Converting to metres: 500 ÷ 100 = 5 m. 500 m comes from forgetting to convert the 500 cm into metres at all. 50 m comes from dividing by 10 instead of 100 when converting centimetres to metres: 500 ÷ 10 = 50. 0.5 m comes from dividing by 1000 instead of 100, as if converting to kilometres instead of metres: 500 ÷ 1000 = 0.5.
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