POP

C1.12 Rates

Image · 914 × 270

01 — Explanation

What is a rate, and how does the speed formula triangle work?

A rate compares two quantities in different units — the word "per" always signals division. The most common rate in this topic is speed:

        D
      -----
     | S T |
      -----

Cover up the quantity you want to find:

\text{Speed} = \frac{\text{Distance}}{\text{Time}}, \qquad \text{Distance} = \text{Speed} \times \text{Time}, \qquad \text{Time} = \frac{\text{Distance}}{\text{Speed}}

02 — Explanation

How do I convert minutes into decimal hours for use in the formula?

Since time runs in base 60, divide the minutes by $60$ :

15 \text{ min} = 0.25\text{ h}, \quad 30 \text{ min} = 0.5\text{ h}, \quad 40 \text{ min} = 0.6\overline{6}\text{ h}

Common Slip

writing $1$ hour $30$ minutes as $1.30$ hours. It should be $1.5$ hours — the $30$ minutes must be divided by $60$ first, not written as if it were a decimal already.

Example: a cyclist rides $54\text{ km}$ in $1$ hour $30$ minutes. Her average speed is:

\text{Speed} = \frac{54}{1.5} = 36 \text{ km/h}

03 — Explanation

How do I find the average speed of a journey with more than one stage?

Average speed is always $\dfrac{\text{total distance}}{\text{total time}}$ — you cannot just average the individual speeds together.

Example: a car travels $80\text{ km}$ at $40\text{ km/h}$ , then $60\text{ km}$ at $30\text{ km/h}$ . Find the average speed for the whole journey.

Time for stage 1: $\dfrac{80}{40} = 2\text{ h}$ .
Time for stage 2: $\dfrac{60}{30} = 2\text{ h}$ .
Total distance $= 80+60=140\text{ km}$ ; total time $= 2+2=4\text{ h}$ .
Average speed $= \dfrac{140}{4} = 35 \text{ km/h}$ .

Exam Tip

here, averaging $40$ and $30$ directly also happens to give $35$ — but that's a coincidence caused by equal times for each stage. Always use the total distance ÷ total time method, since it isn't reliable otherwise.

04 — Explanation

How do rates like density work, using the same triangle method?

Any rate that links two quantities with a "per" relationship can use the same cover-up triangle idea as speed. Density is one of the most common examples:

\text{Density} = \frac{\text{Mass}}{\text{Volume}}, \qquad \text{Mass} = \text{Density} \times \text{Volume}, \qquad \text{Volume} = \frac{\text{Mass}}{\text{Density}}

Example: a metal block has a mass of $270\text{ g}$ and a volume of $30\text{ cm}^3$ . Find its density, then find the mass of a second block of the same metal with volume $50\text{ cm}^3$ .

Density of block 1: $\dfrac{270}{30}=9\text{ g/cm}^3$ .
Since block 2 is the same metal, it has the same density.
Mass of block 2: $9 \times 50 = 450\text{ g}$ .

The required formula will always be given in the question if it isn't speed, but you're still expected to know how to rearrange and substitute into it — exactly as with the speed triangle.

05 — Common mistakes

Common mistakes to avoid

Writing minutes as a decimal directly, e.g. $1$ hour $30$ min as $1.30$ instead of $1.5$ .
Averaging two speeds by adding and halving them rather than using total distance over total time.
Forgetting to convert km/h to m/s (divide by $3.6$ ) or m/s to km/h (multiply by $3.6$ ) when units in the question don't match.