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

1 km

HintMultiply first, then convert centimetres to kilometres.

The why4 × 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

HintOne kilometre is 100,000 cm.

The why100,000 ÷ 50,000 = 2. This is why the grid squares on a 1 : 50,000 map are 2 cm wide.

24 cm

HintConvert the walk to centimetres, then divide by the scale number.

The why6 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

HintAsk on which sheet a given field is drawn bigger.

The whyAt 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

HintShare the mass among the cubic centimetres.

The whyDensity = mass ÷ volume = 240 ÷ 30 = 8. Each cubic centimetre of the block has a mass of 8 g.

g/cm³ (grams per cubic centimetre)

HintThe word 'per' in its name shows a division.

The whyDensity 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

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

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

40 g

HintEach cubic centimetre has a mass of 0.8 g.

The whyMass = density × volume = 0.8 × 50 = 40 g.

€60

HintEach pound becomes a bit more than one euro.

The why50 × 1.20 = 60. The rate tells you the value of one pound, so multiply by the number of pounds.

£25

HintGoing back to pounds, you undo the multiplication.

The why30 ÷ 1.20 = 25. Check: £25 × 1.20 = €30.

Divide

HintYou should end up with fewer pounds than dollars.

The whyMultiplying 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

HintAsk which is worth more: one of ours or one of theirs.

The whyA 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)

HintWhen one quantity is zero, so is the other.

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

£15

HintFind the cost of one kilogram from the point given.

The why£6 ÷ 4 = £1.50 per kilogram, and 10 × £1.50 = £15.

No — the graph does not start at the origin

HintThink about the cost of a journey of zero miles.

The whyThe 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

HintFind what one unit of x is worth.

The why12 ÷ 3 = 4, so y is always 4 times x, and 4 × 5 = 20.