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IMAT Logical Reasoning · Numerical Reasoning

Averages, Conversions, Clocks and Calendars

Work with totals rather than averages, convert every quantity to one unit before calculating, and count time in minutes and days of the week in sevens.

All Numerical Reasoning notes

Why this matters

Costs from tariffs and energy prices, painting a room, repeating patrols, birthdays and delivery rounds appear across the Cambridge papers. They are rarely hard, but a slip with units or with hours and minutes gives one of the wrong options.

Concept 1 of 3: Mean and weighted mean in word problems

An average hides a total. Multiply the mean by the number of items and you get back the total, and totals can be added, subtracted and compared. So almost every average question becomes: find the old total, find the new total, and look at the difference.

Definition

  • Mean = total ÷ number of items, so total = mean × number.
  • A missing value is the new total minus the old total.
  • Combining groups of different sizes gives a weighted mean: each group's mean counts in proportion to its size. Averaging the two means is only right when the groups are equal in size.

Weighted mean

xˉ=n1xˉ1+n2xˉ2n1+n2\bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}
  • n1,n2n_1, n_2the sizes of the two groups
  • xˉ1,xˉ2\bar{x}_1, \bar{x}_2the mean of each group

Worked example

One class of 20 students has a mean mark of 62, and another class of 30 students has a mean of 72. What is the mean of all 50 students? Separately, a student's mean over four tests is 70. What must she score on the fifth test to raise her mean to 74?
Practice this conceptself-check · 3 quick reps

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.

Concept 2 of 3: Unit and currency conversions

A conversion factor is a fraction equal to 1, such as 1000 g over 1 kg. Multiplying by it changes the units but not the amount. Write the units next to every number and cancel them like letters; if the units left at the end are not the ones you want, the factor is upside down.

Definition

  • Multiply by a factor that cancels the old unit and leaves the new one.
  • Area factors are squared and volume factors are cubed: 1 m2=10 000 cm21\ \text{m}^2 = 10\,000\ \text{cm}^2, 1 m3=1000 L1\ \text{m}^3 = 1000\ \text{L}, 1 L=1000 cm31\ \text{L} = 1000\ \text{cm}^3.
  • Energy cost: power in kW × time in hours = energy in kWh; multiply by the price per kWh.
  • Currency: if €1 buys rr dollars, euros to dollars is × rr, dollars to euros is ÷ rr.

Energy used by an appliance

E (kWh)=P (kW)×t (h)E\,(\text{kWh}) = P\,(\text{kW}) \times t\,(\text{h})
  • PPpower; 1 kW = 1000 W
  • tttime switched on, in hours

Worked example

A 2,000 W heater runs for 90 minutes a day for 30 days. Electricity costs €0.25 per kWh. What does it cost to run?
Practice this conceptself-check · 4 quick reps

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.

Concept 3 of 3: Time, clocks, calendars and repeating cycles

Time is not decimal: there are 60 minutes in an hour, 24 hours in a day and 7 days in a week. The safe method is to convert to the smallest unit (minutes or days), calculate, and convert back. Things that repeat at different intervals line up again after the lowest common multiple of the intervals.

Definition

  • Clock time: turn times into minutes after midnight, subtract, convert back. A journey past midnight adds 24 h (1440 min) to the arrival.
  • Days of the week repeat every 7 days: nn days later is the remainder of n÷7n \div 7 days later in the week.
  • Month lengths: 30 days for April, June, September, November; 28 or 29 for February; 31 for the rest.
  • Leap years are divisible by 4, except century years, which must be divisible by 400 (2000 was, 2100 is not).
  • Repeating cycles coincide again after the lowest common multiple (LCM) of their periods.
  • Counting days: from day 15 to day 20 is 5 days later but covers 6 dates if both ends count.

Day of the week after n days

shift=n mod 7\text{shift} = n \bmod 7
  • n mod 7n \bmod 7the remainder when nn is divided by 7

Worked example

Two shuttle buses leave a station together at 07:00. One returns to the station every 12 minutes, the other every 18 minutes. When are they next at the station together? Separately, if 1 March is a Tuesday, what day is 1 April?
Practice this conceptself-check · 4 quick reps

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.

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 (3)

Watch out for (5)

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