Compound Measures

Divide the mass by the volume.

1. Density is mass divided by volume.
2. 84\div28=3, so the density is 3\,\text{g}/\text{cm}^3.

Mass is density multiplied by volume.

1. Mass is density multiplied by volume.
2. 3.5\times24=84, so the mass is 84\,\text{g}.

Divide the distance by the time.

1. Average speed is distance divided by time.
2. 540\div90=6, so the speed is 6\,\text{m}/\text{s}.

Convert the time to seconds before dividing the distance by the time.

1. 2.5 minutes is 2.5\times60=150 seconds.
2. The average speed is 1500\div150=10\,\text{m}/\text{s}.

Convert the area to square metres, then divide force by area.

1. 800\,\text{cm}^2=0.08\,\text{m}^2.
2. Pressure is force divided by area, so 240\div0.08=3000\,\text{N}/\text{m}^2.

Divide the volume by the time.

1. Mean flow rate is volume divided by time.
2. 18\div3=6, so the flow rate is 6 litres per minute.

Change the mass to grams so that it matches the density, then rearrange density = mass ÷ volume.

1. The density is in g/cm^3, so change the mass to grams: 2.1\,\text{kg}=2100\,\text{g}.
2. Density is mass divided by volume, so volume is mass divided by density.
3. 2100\div8.4=250, so the volume is 250\,\text{cm}^3.

Write the time as a fraction of an hour before dividing.

1. The speed must be in km per hour, so write the time in hours: 40 minutes is \frac{40}{60}=\frac23 of an hour.
2. Average speed is distance divided by time: 12\div\frac23=12\times\frac32=18.
3. The average speed is 18\,\text{km}/\text{h}.

Time is distance divided by speed.

1. Time is distance divided by speed.
2. 84\div56=1.5, so the journey takes 1.5 hours.

Convert the time to hours, then multiply speed by time.

1. 35 minutes is 35\div60=\frac{7}{12} hours.
2. Distance is speed multiplied by time: 72\times\frac{7}{12}=42\,\text{km}.

Calculate force divided by area for each press.

1. Press A exerts a pressure of 500\div0.10=5000\,\text{N}/\text{m}^2.
2. Press B exerts a pressure of 600\div0.15=4000\,\text{N}/\text{m}^2.
3. Since 5000>4000, press A exerts the greater pressure.

Rearrange pressure = force divided by area to find the force.

1. Pressure equals force divided by area, so force equals pressure multiplied by area.
2. 7500\times0.08=600, so the force is 600\,\text{N}.

Find the time for each part, then divide total distance by total time.

1. The first part takes 30\div60=0.5 hours and the second takes 30\div30=1 hour.
2. The whole journey is 60\,\text{km} in 1.5 hours.
3. The average speed is 60\div1.5=40\,\text{km}/\text{h}.

Convert kilometres to metres and hours to seconds.

1. 90\,\text{km}/\text{h}=90\times1000\div3600\,\text{m}/\text{s}.
2. This is 25\,\text{m}/\text{s}.

Convert both lengths to metres, then divide the force by the base area.

1. The base measures 0.4\,\text{m} by 0.3\,\text{m}, so its area is 0.12\,\text{m}^2.
2. Pressure is force divided by area.
3. 720\div0.12=6000\,\text{N}/\text{m}^2.

Find the volume in cubic centimetres and convert the mass to grams.

1. The length is 120\,\text{cm}, so the volume is 120\times5\times4=2400\,\text{cm}^3.
2. The mass is 18.72\,\text{kg}=18720\,\text{g}.
3. Density is 18720\div2400=7.8\,\text{g}/\text{cm}^3.

Find the volume of water needed in litres before using the flow rate.

1. The capacity is 50\times40\times30=60000 cm^3.
2. Since 1000 cm^3 is 1 litre, the capacity is 60 litres. Three-quarters of this is 45 litres.
3. Time is volume divided by flow rate: 45\div2.5=18 minutes.

Find the total mass of the mixture, then subtract the mass of liquid A.

1. The mixture has mass 1.8\times250=450\,\text{g}.
2. Liquid A has mass 2.4\times100=240\,\text{g}, so liquid B has mass 450-240=210\,\text{g}.
3. Liquid B has volume 250-100=150\,\text{cm}^3, so its density is 210\div150=1.4\,\text{g}/\text{cm}^3.

Average density is the total mass divided by the total volume.

1. Volumes: the cube is 4^3=64\,\text{cm}^3 and the cuboid is 4\times4\times6=96\,\text{cm}^3, so the total volume is 160\,\text{cm}^3.
2. Mass is density multiplied by volume. The cube has mass 9\times64=576\,\text{g} and the cuboid has mass 0.75\times96=72\,\text{g}.
3. The total mass is 576+72=648\,\text{g}.
4. Average density =\frac{\text{total mass}}{\text{total volume}}=\frac{648}{160}=4.05\,\text{g}/\text{cm}^3.
5. Averaging the two densities, (9+0.75)\div2, is wrong because the two parts have different volumes.

Write a multiplier for the force and one for the area, then use pressure = force ÷ area.

1. Pressure is force divided by area.
2. The force is multiplied by 1.2. The area is multiplied by 0.75.
3. So the pressure is multiplied by 1.2\div0.75=1.6.
4. The new pressure is 160\% of the old pressure, an increase of 60\%. (Adding 20\% and 25\% gives the wrong answer: the area change divides the pressure, it doesn't add to it.)