Ratio and proportion, Year 8: maps, density, currency and graphs

1 km

提示Multiply first, then convert centimetres to kilometres.

为什么4 × 25,000 = 100,000 cm. That is 1,000 m, which is 1 km. On this scale, 4 cm on the map is always 1 km.

2

提示One kilometre is 100,000 cm.

为什么100,000 ÷ 50,000 = 2. This is why the grid squares on a 1 : 50,000 map are 2 cm wide.

24 cm

提示Convert the walk to centimetres, then divide by the scale number.

为什么6 km = 600,000 cm, and 600,000 ÷ 25,000 = 24 cm. Or use 4 cm for each kilometre: 6 × 4 = 24.

The 1 : 25,000 map

提示Ask on which sheet a given field is drawn bigger.

为什么At 1 : 25,000, a kilometre takes 4 cm of paper; at 1 : 50,000 it takes only 2 cm. More paper for each kilometre leaves room for more detail.

8 g/cm³

提示Share the mass among the cubic centimetres.

为什么Density = mass ÷ volume = 240 ÷ 30 = 8. Each cubic centimetre of the block has a mass of 8 g.

g/cm³ (grams per cubic centimetre)

提示The word 'per' in its name shows a division.

为什么Density is a rate: it tells you how many grams there are in each cubic centimetre. The unit comes from dividing one unit by the other.

Block A

提示The volumes are equal, so compare what is packed into each.

为什么With equal volumes, the block with more mass has more grams in every cubic centimetre, so it is the denser one.

40 g

提示Each cubic centimetre has a mass of 0.8 g.

为什么Mass = density × volume = 0.8 × 50 = 40 g.

€60

提示Each pound becomes a bit more than one euro.

为什么50 × 1.20 = 60. The rate tells you the value of one pound, so multiply by the number of pounds.

£25

提示Going back to pounds, you undo the multiplication.

为什么30 ÷ 1.20 = 25. Check: £25 × 1.20 = €30.

Divide

提示You should end up with fewer pounds than dollars.

为什么Multiplying by 1.25 turns pounds into dollars, so dividing by 1.25 goes the other way.

The number of pounds must be smaller than the number of euros

提示Ask which is worth more: one of ours or one of theirs.

为什么A pound is worth more than a euro, so you need fewer of them. The correct value is 240 ÷ 1.20 = £200; the tourist multiplied.

The origin, (0, 0)

提示When one quantity is zero, so is the other.

为什么Nothing bought costs nothing; no time travelled covers no distance. So the line has to start where both quantities are zero.

£15

提示Find the cost of one kilogram from the point given.

为什么£6 ÷ 4 = £1.50 per kilogram, and 10 × £1.50 = £15.

No — the graph does not start at the origin

提示Think about the cost of a journey of zero miles.

为什么The graph is a straight line, but it starts at £3. Doubling the miles does not double the fare, so it is not direct proportion.

20

提示Find what one unit of x is worth.

为什么12 ÷ 3 = 4, so y is always 4 times x, and 4 × 5 = 20.