SQA National 5 Applications of Mathematics

Notation

Applications of Mathematics problems are written in words and symbols, and the first job is to read the symbols correctly.

When a calculation has several operations, brackets are done first, then multiplication and division from left to right, then addition and subtraction from left to right. So $4 + 6 \times 3 = 22$, since the multiplication comes first, but $(4 + 6) \times 3 = 30$.

A simple formula is a rule written with letters. To use it, replace each letter with its value and work out the result. If the cost in pounds of hiring a van for $d$ days is $C = 25 + 40d$, then for 3 days

so the hire costs £145. The multiplication is done before the addition.

The inequality symbols compare amounts. A result of 7.4 is less than 8, written $7.4 < 8$, and a mass of 12 kg is greater than 9.5 kg, written $12 > 9.5$. The open end of the symbol faces the larger number.

Units

A number in a real situation is almost always attached to a unit, and the unit has to be written with the answer. The units used most often are below.

Choose the unit that suits the size of the thing being measured. The length of a pencil is given in centimetres, the distance between two towns in kilometres, and the mass of a bag of flour in grams or kilograms. An answer such as 0.004 km is correct, but 4 m is the sensible way to say it.

Converting between units

To go from a larger unit to a smaller one, multiply, because there are more of the small units. To go the other way, divide.

Changing 3.6 km to metres: $3.6 \times 1000 = 3600$ m.

Changing 450 g to kilograms: $450 \div 1000 = 0.45$ kg.

Changing 2.5 litres to millilitres: $2.5 \times 1000 = 2500$ ml.

Money needs care with the decimal point. 35p is £0.35, not £0.035, and £4.50 is 450p. A money answer is written to two decimal places, so £6.5 should be £6.50.

Time does not work in tens. A time of 2.5 hours is 2 hours 30 minutes, since 0.5 of an hour is 30 minutes, but 2.25 hours is 2 hours 15 minutes, and 2 hours 20 minutes is $2 + \dfrac{20}{60} = 2.33$ hours, not 2.20 hours.

Reading a scale on an instrument

Rulers, protractors, measuring jugs, kitchen and bathroom scales and thermometers all have a scale with numbered marks and, between them, smaller unnumbered marks. A measurement is recorded to the nearest marked division, including the minor unnumbered ones, and the unit is written with it.

To read a scale:

1. Find two neighbouring numbered marks and subtract to get the amount between them.
2. Count how many equal spaces lie between them.
3. Divide the amount by the number of spaces to find what each minor division is worth.
4. Count on from the lower numbered mark to the pointer or the liquid level, and then read the value of the nearest minor division.

If the numbered marks on a jug are 200 ml and 300 ml and there are 5 spaces between them, each space is $100 \div 5 = 20$ ml. A level that sits on the second mark after 200 ml reads $200 + 2 \times 20 = 240$ ml. If the level falls between two marks, it is recorded as the nearer one.

The same method works on a thermometer. With numbered marks at 20 °C and 30 °C and ten spaces between them, each space is 1 °C, and a column that reaches seven spaces above 20 reads 27 °C. A scale with marks at 0 and 1 kg split into four spaces goes up in 0.25 kg steps, so a pointer just past the third mark reads 0.75 kg.

On a ruler the nearest minor division is usually a millimetre, so a length is recorded to the nearest millimetre, and angles measured with a protractor are recorded to the nearest degree. A measurement read from an instrument is only as accurate as its smallest division, and this matters when it is compared against a limit.