Every card in Ratio and proportion, Year 8: maps, density, currency and graphs
The whole deck, in order — so you can read it through before your child ever sees it.
← Back to Ratio and proportion, Year 8: maps, density, currency and graphs
- On a 1 : 25,000 map, two farms are 4 cm apart. How far apart are they in real life, in kilometres?
1 km
HintMultiply first, then convert centimetres to kilometres.
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.
- On a 1 : 50,000 map, how many centimetres represent 1 km? (number only)
2
HintOne kilometre is 100,000 cm.
Why100,000 ÷ 50,000 = 2. This is why the grid squares on a 1 : 50,000 map are 2 cm wide.
- A walk is 6 km long. How long is it on a 1 : 25,000 map?
24 cm
HintConvert the walk to centimetres, then divide by the scale number.
Why6 km = 600,000 cm, and 600,000 ÷ 25,000 = 24 cm. Or use 4 cm for each kilometre: 6 × 4 = 24.
- Which shows more detail of a small area: a 1 : 25,000 map or a 1 : 50,000 map?
The 1 : 25,000 map
HintAsk on which sheet a given field is drawn bigger.
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.
- A block has a mass of 240 g and a volume of 30 cm³. What is its density?
8 g/cm³
HintShare the mass among the cubic centimetres.
WhyDensity = mass ÷ volume = 240 ÷ 30 = 8. Each cubic centimetre of the block has a mass of 8 g.
- What is the unit of density when mass is measured in grams and volume in cm³?
g/cm³ (grams per cubic centimetre)
HintThe word 'per' in its name shows a division.
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.
- Two blocks are the same size. Block A has a mass of 500 g and block B has a mass of 200 g. Which is made of the denser material?
Block A
HintThe volumes are equal, so compare what is packed into each.
WhyWith equal volumes, the block with more mass has more grams in every cubic centimetre, so it is the denser one.
- A liquid has a density of 0.8 g/cm³. What is the mass of 50 cm³ of it?
40 g
HintEach cubic centimetre has a mass of 0.8 g.
WhyMass = density × volume = 0.8 × 50 = 40 g.
- The exchange rate is £1 = €1.20. How many euros do you get for £50?
€60
HintEach pound becomes a bit more than one euro.
Why50 × 1.20 = 60. The rate tells you the value of one pound, so multiply by the number of pounds.
- The exchange rate is £1 = €1.20. How many pounds is €30 worth?
£25
HintGoing back to pounds, you undo the multiplication.
Why30 ÷ 1.20 = 25. Check: £25 × 1.20 = €30.
- The rate is £1 = $1.25. To change dollars into pounds, do you multiply or divide by 1.25?
Divide
HintYou should end up with fewer pounds than dollars.
WhyMultiplying by 1.25 turns pounds into dollars, so dividing by 1.25 goes the other way.
- At £1 = €1.20, a tourist says that €240 is worth £288. Without an exact calculation, how do you know this is wrong?
The number of pounds must be smaller than the number of euros
HintAsk which is worth more: one of ours or one of theirs.
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 graph of two quantities in direct proportion is a straight line through which point?
The origin, (0, 0)
HintWhen one quantity is zero, so is the other.
WhyNothing bought costs nothing; no time travelled covers no distance. So the line has to start where both quantities are zero.
- A straight-line graph of cost against mass passes through (0, 0) and (4 kg, £6). What is the cost of 10 kg?
£15
HintFind the cost of one kilogram from the point given.
Why£6 ÷ 4 = £1.50 per kilogram, and 10 × £1.50 = £15.
- A taxi charges £3 plus £2 a mile. Is the fare directly proportional to the distance?
No — the graph does not start at the origin
HintThink about the cost of a journey of zero miles.
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.
- y is directly proportional to x, and y = 12 when x = 3. What is y when x = 5?
20
HintFind what one unit of x is worth.
Why12 ÷ 3 = 4, so y is always 4 times x, and 4 × 5 = 20.
1★ KS3-MATH-RAT-0056
2★ KS3-MATH-RAT-0057
3★ KS3-MATH-RAT-0058
4★ KS3-MATH-RAT-0059
5★ KS3-MATH-RAT-0060
6★ KS3-MATH-RAT-0061
7★ KS3-MATH-RAT-0062
8★ KS3-MATH-RAT-0063
9★ KS3-MATH-RAT-0064
10★ KS3-MATH-RAT-0065
11★ KS3-MATH-RAT-0066
12★ KS3-MATH-RAT-0067
13★ KS3-MATH-RAT-0068
14★ KS3-MATH-RAT-0069
15★ KS3-MATH-RAT-0070
16★ KS3-MATH-RAT-0071
Keep what you learn
Here, nothing is saved. In your child’s own sky every card is scheduled — it comes back just before they’d forget it — and the professor who wrote it is one tap away.