Numerical Reasoning — IMAT Formulas and Traps

Turning a Story into Arithmetic

Linked relations: substituting one into another

Chaining two relations

a = kb \ \text{and}\ b = mc \;\Rightarrow\; a = kmc, \quad c = \frac{a}{km}

$k, m$ the multipliers in each relation
$a, b, c$ the linked quantities; $b$ is the shared letter that disappears

Common traps

Solving for the wrong quantity

If the question asks for the teacher tickets and your letter stands for the student tickets, the number you find is a correct answer to a different question. IMAT puts that number among the options. Re-read the last line of the question before choosing.

"5 less than x" is x − 5

The order in the sentence is the reverse of the order in the algebra: "5 less than

x

" is

x - 5

, not

5 - x

. In the same way, "A has 3 more than B" is

A = B + 3

.

Chained multipliers multiply

If

a = 4b

and

b = 5c

, then

a = 20c

, not

9c

. Adding the two multipliers is the most common wrong option. Test any answer with a small number before you choose it.

Undo the steps in reverse order

If the story says "take half, then subtract 2", going backwards you add 2 first and double second. Doubling first and then adding 2 gives a different, wrong starting value, and that value is often one of the options.

Percentages: Change, Reverse and Parts of Parts

Learn this subtopic in the notes

Percentage change and the multiplier

Percentage change

\text{percentage change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\%

$\text{original}$ the value before the change (the base of the percentage)
$\text{new}$ the value after the change

Successive changes and reverse percentages

Successive changes and the original value

\text{final} = \text{original} \times m_1 \times m_2 \qquad \text{original} = \frac{\text{final}}{m}

$m_1, m_2$ the multipliers of each change, e.g. 0.80 for 20% off
$m$ the overall multiplier from the original to the final value

Percentages of a part, and percentage points

Part of a part

\text{fraction of the whole} = \frac{p_1}{100} \times \frac{p_2}{100}

$p_1$ the percentage that picks out the first group
$p_2$ the percentage of that group that is wanted

Common traps

Percentage change divides by the original

A rise from 12,500 to 13,250 is 750 out of 12,500, which is 6%. Dividing by the new value, 13,250, gives about 5.7%, and that wrong value is usually among the options.

Successive percentages do not add

20% off followed by 15% off is a 32% discount, not 35%, because the second discount is taken from a smaller price. In the same way, two 20% rises make 44%, not 40%.

To find the original, divide by the multiplier

If a price after an 8% rise is €2,160, subtracting 8% of €2,160 gives €1,987.20, which is wrong. The 8% was 8% of the old price, so divide:

2160 / 1.08 = 2000

.

Percentage points are not percent

A rate moving from 8% to 6% falls by 2 percentage points, but by 25% of its old value. An option saying "it fell by 2%" uses the wrong unit and is wrong.

"Of the rest" means of the remainder, not of the whole

If 40% take the bus and 25% of the rest cycle, the cyclists are 25% of 60%, which is 15% of the school, not 25%.

Ratio, Proportion, Speed and Work

Learn this subtopic in the notes

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}

$r_i$ the number of parts for share $i$

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

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

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

Work and flow rates: adding rates, not times

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

Common traps

2 : 5 means 2 parts out of 7

In a 2 : 5 mixture, the first ingredient is

\dfrac{2}{7}

of the total. Using

\dfrac{2}{5}

compares one ingredient with the other, not with the whole mixture.

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.

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.

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.

Averages, Conversions, Clocks and Calendars

Learn this subtopic in the notes

Mean and weighted mean in word problems

Weighted mean

\bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}

$n_1, n_2$ the sizes of the two groups
$\bar{x}_1, \bar{x}_2$ the mean of each group

Unit and currency conversions

Energy used by an appliance

E\,(\text{kWh}) = P\,(\text{kW}) \times t\,(\text{h})

$P$ power; 1 kW = 1000 W
$t$ time switched on, in hours

Time, clocks, calendars and repeating cycles

Day of the week after n days

\text{shift} = n \bmod 7

$n \bmod 7$ the remainder when $n$ is divided by 7

Common traps

Do not average two averages of unequal groups

A class of 20 with mean 62 and a class of 30 with mean 72 have a combined mean of 68, not 67. The larger group pulls the combined mean towards its own mean. Always go back to the totals.

Area and volume factors are squared and cubed

1 m = 100 cm, but 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³. Using 100 for an area conversion gives an answer 100 times too small or too large.

Exchange rates work in one direction at a time

If €1 buys $1.08, then converting dollars back to euros means dividing by 1.08. Multiplying again makes the money grow on the way back, which cannot happen at the same rate.

Hours are not decimals

1.5 h is 1 h 30 min, not 1 h 50 min, and 2 h 50 min is about 2.83 h, not 2.5 h. Convert to minutes before adding or dividing times.

Counting both ends

A fence 30 m long with a post every 3 m has 10 gaps but 11 posts. A stay from the 3rd to the 7th is 4 nights but 5 dates. Decide whether the question counts gaps or end points.

Counting Systematically, Overlaps and Chance

Learn this subtopic in the notes

Systematic listing and the multiplication rule

Choosing r items from n

\text{ordered: } \frac{n!}{(n-r)!} \qquad \text{unordered: } \binom{n}{r} = \frac{n!}{r!\,(n-r)!}

$n$ number of items available
$r$ number chosen

Overlapping groups and Venn reasoning

Two overlapping groups

n(A \cup B) = n(A) + n(B) - n(A \cap B)

$A \cup B$ in A or B or both
$A \cap B$ in both A and B

Probability in context

Two events in turn

P(A \text{ then } B) = P(A) \times P(B \text{ given } A)

$P(B \text{ given } A)$ probability of B once A has happened; equals $P(B)$ if they are independent

Common traps

Does order matter?

A code 123 differs from 321, so order matters. A pair of friends AB is the same as BA, so order does not. Using the ordered count for a selection doubles (or more) the true answer.

Adding two groups counts the overlap twice

45 with one condition and 30 with another do not make 75 people with a condition if some have both. Subtract the overlap once:

n(A) + n(B) - n(A \cap B)

.

Without replacement, the second fraction changes

After one red counter is removed, there is one red fewer and one counter fewer. Squaring the first probability treats the draws as if the counter had been put back.

A positive test is not the same as having the condition

When a condition is rare, most positive results can come from the large healthy group. Count people in each group out of a round total before answering.

Repeated Growth, Half-Life and Checking by Estimate

Learn this subtopic in the notes

Repeated growth, doubling and half-life

Growth and decay

N = N_0\, r^{\,n} \qquad N = N_0 \left(\tfrac{1}{2}\right)^{t/T}

$N_0$ starting amount
$r, n$ factor per step, number of steps
$T$ half-life, in the same unit as the time $t$

Common traps

Repeated percentages compound

10% a year for 2 years is a 21% rise, not 20%, because the second 10% is taken on the larger amount. Use the multiplier raised to the number of years.

Two half-lives leave a quarter, not nothing

After one half-life half remains; after two, a quarter; after three, an eighth. An option saying the substance is gone after two half-lives treats the halving as a straight-line fall.

"How many are needed" rounds up

If a wall needs 4.2 litres of paint and paint comes in 1 litre tins, you must buy 5 tins. Rounding to the nearest whole number gives 4, which leaves part of the wall unpainted. "How many fit" rounds down instead.

Numerical Reasoning formulas

Turning a Story into Arithmetic

Linked relations: substituting one into another

Common traps

Percentages: Change, Reverse and Parts of Parts

Percentage change and the multiplier

Successive changes and reverse percentages

Percentages of a part, and percentage points

Common traps

Ratio, Proportion, Speed and Work

Sharing a quantity in a given ratio

Direct and inverse proportion, and value for money

Speed, distance and time, and average speed

Work and flow rates: adding rates, not times

Common traps

Averages, Conversions, Clocks and Calendars

Mean and weighted mean in word problems

Unit and currency conversions

Time, clocks, calendars and repeating cycles

Common traps

Counting Systematically, Overlaps and Chance

Systematic listing and the multiplication rule

Overlapping groups and Venn reasoning

Probability in context

Common traps

Repeated Growth, Half-Life and Checking by Estimate

Repeated growth, doubling and half-life

Common traps

More IMAT Logical Reasoning and Problem Solving formula sheets