Rates, ratios & proportions · Ages 13–14
Density, pressure and exchange rates
The unit says what to do: g/cm³ is g ÷ cm³.
How to do it
Method
The unit says what to do: g/cm³ is g ÷ cm³
540 ÷ 200 = 2.7 (g/cm³)
Another way
The formula triangle
Mass = density × volume = 2.7 × 200 = 540
In full
Many everyday quantities are rates: one quantity divided by another, the unit written "AB". Density = mass ÷ volume (g/cm³), pressure = force ÷ area (N/m²), population density = population ÷ area (people/km²), flow rate = volume ÷ time (L/minute). One way: let the unit say whether to multiply or divide. g/cm³ is g ÷ cm³: a 540 g block of metal with a volume of 200 cm³ has a density of 540 ÷ 200 = 2.7 g/cm³; the other way, mass = density × volume = 2.7 × 200 = 540 g, and volume = mass ÷ density. Another: the formula triangle, mass on top, density and volume below; cover the one you want, and one above the other is a division, side by side a multiplication. Put the units in step first: 1 m³ = 1000 L, 1 L = 1000 cm³, 1 m² = 10000 cm². An exchange rate is a rate too: if 100 Hong Kong dollars buy 1850 yen one day, 500 buy 1850 × 5 = 9250 yen; the other way, 3700 yen = 3700 ÷ 1850 × 100 = 200 Hong Kong dollars. Or find what 1 Hong Kong dollar buys first, 18.5 yen, then multiply or divide. To compare a price abroad with one at home, turn both into the same currency first.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A district has an area of 33 km² and a population of 818400. Its population density =
Show the answer and working
Answer: 24800 people/km²
Population density = population ÷ area = 818400 ÷ 33 = 24800 people/km², that is 24800 people for each km² on average.
Example 2
A hose lets in 25 L of water a minute. Filling an empty tank that holds 0.8 m³ takes 32 minutes.
True or false?
Show the answer and working
Answer: True
1 m³ = 1000 L, so 0.8 m³ = 800 L; time = volume ÷ flow rate = 800 ÷ 25 = 32 minutes.
Example 3
2.7 g/cm³ = kg/m³
Show the answer and working
Answer: 2700
1 m³ = 1000000 cm³ and 1 kg = 1000 g: 2.7 g/cm³ = 2.7 × 1000000 ÷ 1000 = 2700 kg/m³, that is, times 1000.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.One day, 1 pound buys 10.2 Hong Kong dollars. How many Hong Kong dollars do 450 pounds buy?
- A. 460.2 Hong Kong dollars
- B. 459 Hong Kong dollars
- C. About 441.2 Hong Kong dollars
- D. 4590 Hong Kong dollars
Choose the right answer
Answer:
2.A block has a mass of 2938 g and a volume of 260 cm³. Its density =
Answer:
3.A box weighs 2925 N and its base has an area of 0.5 m². Standing on the floor, the pressure it puts on the floor =
Answer:
4.The same headphones cost 370 Hong Kong dollars in Hong Kong and 8299 yen in Japan. One day, 100 Hong Kong dollars buy 1930 yen. Leaving other costs aside, where are they cheaper?
- A. In Hong Kong
- B. The same in both
- C. In Japan
- D. They can't be compared
Choose the right answer
Answer:
5.8.9 g/cm³ = kg/m³
Answer:
All the answers
- 1. D. 4590 Hong Kong dollars
- 2. 11.3 g/cm³
- 3. 5850 N/m²
- 4. A. In Hong Kong
- 5. 8900
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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