IMAT Logical Reasoning · Numerical Reasoning

Ratio, Proportion, Speed and Work

Share in parts, scale with the unit method, decide whether two quantities grow together or one shrinks as the other grows, and add rates rather than times.

All Numerical Reasoning notes

Why this matters

Ratio sharing, best-value comparisons and journeys appear in most Cambridge papers: trains through tunnels, boats meeting on a crossing, average speeds and how many workers a job needs.

Concept 2 of 4: Direct and inverse proportion, and value for money

Ask one question first: if one quantity doubles, does the other double or halve? Twice as many kilograms cost twice as much (direct). Twice as many workers finish in half the time (inverse). For direct proportion find the value for one unit; for inverse proportion keep the product fixed. Value for money is direct proportion: compare the price of one common unit.

Definition

Direct proportion: $y = kx$ . The ratio $y/x$ stays the same. Use the unit method: find the value for 1, then scale.
Inverse proportion: $y = k/x$ . The product $xy$ stays the same (for example worker-hours or pump-hours).
Best value: convert every offer to the price of the same amount (per kg, per litre, per wash) before comparing. A bigger pack is not automatically cheaper.

Direct and inverse proportion

\text{direct: } \frac{y_1}{x_1} = \frac{y_2}{x_2} \qquad \text{inverse: } x_1 y_1 = x_2 y_2

$x, y$ the two related quantities, before (1) and after (2)

Worked example

Six identical pumps empty a pool in 10 hours. How long would four pumps take? In the same town, 5 kg of rice costs €7.50. What do 8 kg cost?

Practice this conceptself-check · 3 quick reps

More workers means less time

Workers and time are in inverse proportion. Scaling the time up with the number of workers (more workers, longer job) is the wrong kind of proportion. Check by asking what happens if the number of workers doubles.

Concept 3 of 4: Speed, distance and time, and average speed

Speed is distance per unit time, so any one of the three follows from the other two. Average speed for a whole journey is the total distance divided by the total time. If you spend longer at the slow speed, the average is pulled towards the slow speed, so it is not the simple mean of the two speeds. When two things move towards each other, the gap closes at the sum of their speeds.

Definition

$v = \dfrac{d}{t}$ , $d = vt$ , $t = \dfrac{d}{v}$ . Keep units consistent: km with hours, m with seconds.
$1\ \text{m/s} = 3.6\ \text{km/h}$ .
Average speed = total distance ÷ total time, including stops.
Relative speed: moving towards each other, the gap closes at $v_1 + v_2$ ; moving the same way, at $v_1 - v_2$ .

Speed and average speed

v = \frac{d}{t} \qquad \bar{v} = \frac{d_{\text{total}}}{t_{\text{total}}}

$d$ distance
$t$ time taken
$\bar{v}$ average speed for the whole journey

Worked example

A cyclist rides 30 km at 20 km/h and then another 30 km at 12 km/h. What is the average speed for the whole ride?

Practice this conceptself-check · 4 quick reps

Average speed is not the average of the speeds

Equal distances at 20 and 12 km/h give an average of 15 km/h, not 16. Only equal TIMES at each speed give the simple mean. Always divide the total distance by the total time.

Concept 4 of 4: Work and flow rates: adding rates, not times

If a tap fills a tank in 6 hours, it fills one sixth of the tank each hour. That fraction per hour is its rate. Two taps working together add their rates, and a drain subtracts its rate. Once you have the combined rate, the time is one whole job divided by that rate.

Definition

Rate of a worker, tap or machine = (one job) ÷ (time it takes alone).
Working together, rates add; something working against the job (a drain, a leak) has a negative rate.
Time together = $\dfrac{1}{\text{combined rate}}$ . It is always shorter than the fastest worker's time alone.
Flow problems with volumes work the same way: litres per minute add.

Two workers together

\frac{1}{T} = \frac{1}{t_1} + \frac{1}{t_2}

$t_1, t_2$ time each one needs alone
$T$ time needed working together

Worked example

Tap A fills a tank in 6 hours and tap B fills it in 3 hours. How long do they take together? How long if a drain that would empty the full tank in 4 hours is also open?

Practice this conceptself-check · 3 quick reps

Rates add; times do not

Two printers taking 15 and 10 minutes do not take 25 minutes, or 12.5 minutes, together. Together they must be faster than the faster one alone. Add the fractions of the job done per minute, then invert.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Sharing a quantity in a given ratio
Share in a ratio
$\text{share}_i = \text{total} \times \frac{r_i}{r_1 + r_2 + \dots}$
Direct and inverse proportion, and value for money
Direct and inverse proportion
$\text{direct: } \frac{y_1}{x_1} = \frac{y_2}{x_2} \qquad \text{inverse: } x_1 y_1 = x_2 y_2$
Speed, distance and time, and average speed
Speed and average speed
$v = \frac{d}{t} \qquad \bar{v} = \frac{d_{\text{total}}}{t_{\text{total}}}$
Work and flow rates: adding rates, not times
Two workers together
$\frac{1}{T} = \frac{1}{t_1} + \frac{1}{t_2}$

Watch out for (4)

2 : 5 means 2 parts out of 7
→ Sharing a quantity in a given ratio
More workers means less time
→ Direct and inverse proportion, and value for money
Average speed is not the average of the speeds
→ Speed, distance and time, and average speed
Rates add; times do not
→ Work and flow rates: adding rates, not times

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