p quantity is found by multiplying: $b=a \times c$. - The bottom quantity is found by dividing: $c=\frac{b}{a}$. In formulas shaped like “top divided by bottom”, cover the missing quantity. If the missing quantity is on top, multiply. If it is on the bottom, divide. 1. A stone has mass 84 g and density 3.5 g/cm³. Volume is the bottom quantity, so divide mass by density. $$ \text{volume} = \frac{84}{3.5} = 24 $$ 2. The volume is 24 cm³. 3. A block has force 108 N and pressure 36 N/m². Area is the bottom quantity, so divide force by pressure. $$ \text{area} = \frac{108}{36} = 3 $$ 4. The area is 3 m². ## Time units in speed questions For a speed in **miles per hour**, the time must be in hours. For a speed in **metres per second**, the time must be in seconds. To convert minutes to hours, divide by 60. To convert seconds to minutes, divide by 60. Do not write 4 hours 15 minutes as 4.15 hours. Since 15 minutes is $\frac{15}{60}=0.25$ hours, the correct time is 4.25 hours. 1. A car travels 310 miles in 5 hours 40 minutes. Convert the time into hours. $$ 5+\frac{40}{60}=5.666\ldots $$ 2. Use average speed equals distance divided by time. $$ \text{average speed}=\frac{310}{5.666\ldots}=54.705\ldots $$ 3. To 1 decimal place, the average speed is 54.7 mph. 4. For a separate journey, a driver travels 81 miles at 36 mph. Find the time. $$ \text{time}=\frac{81}{36}=2.25 $$ 5. 2.25 hours is 2 hours 15 minutes, so if the driver leaves at 08:20, they arrive at 10:35. ## Average speed for a whole journey For a whole journey, use: $$ \text{average speed}=\frac{\text{total distance}}{\text{total time}} $$ This is true even if the speed changes during the journey. If a journey has two parts, do not usually add the two speeds and divide by 2. Use total distance divided by total time instead. A van travels 150 miles at 60 mph. It then travels another 90 miles in 2 hours 15 minutes. Find the average speed for the whole journey. 1. Find the time for the first part. $$ \text{time}=\frac{150}{60}=2.5 $$ 2. Convert the second time into hours. $$ 2+\frac{15}{60}=2.25 $$ 3. Add the distances and times. $$ \begin{aligned} \text{total distance} &= 150+90=240 \\ \text{total time} &= 2.5+2.25=4.75 \end{aligned} $$ 4. Find the average speed. $$ \text{average speed}=\frac{240}{4.75}=50.526\ldots $$ 5. The average speed is about 50.5 mph. If two people take different roads, the distance may be different. You can only use the same distance if the question makes it clear they travelled the same journey. ## Mixing liquids and density A **mixture** is made by combining substances. In these questions, work with **mass** and **volume** separately. Do not add densities directly. Density is found after you know the total mass and total volume. For mixed liquids: find masses, find volumes, then use density equals total mass divided by total volume. 120 cm³ of Liquid A has density 1.25 g/cm³. It is mixed with some Liquid B to make Liquid C. Liquid C has mass 240 g and density 1.2 g/cm³. Find the density of Liquid B. 1. Find the total volume of Liquid C. $$ \text{volume of C}=\frac{240}{1.2}=200 $$ 2. Find the mass of Liquid A. $$ \text{mass of A}=1.25 \times 120=150 $$ 3. Subtract to find the volume and mass of Liquid B. $$ \begin{aligned} \text{volume of B} &= 200-120=80 \\ \text{mass of B} &= 240-150=90 \end{aligned} $$ 4. Find the density of L