Maths Skills in Biology (Cambridge (CIE) IGCSE Biology): Revision Note

How to calculate magnification

• The magnification of an object, e.g. when viewed under a microscope, can be calculated as follows:

magnification = image size ÷ actual size

• The image size is the size of the drawing or photograph, as measured with a ruler

• The actual size is the real size of the specimen

• When using the formula in exam questions, it is essential that you convert any units so that they match, e.g. if the actual size of a cell is given in μm, you need to convert the image size into μm as well

  • This can be achieved by measuring the image size in millimetres and then multiplying this value by 1000 (there are 1000 μm in a mm)

• Remember that magnification has no unit, and is usually expressed as, e.g. ×400

• The magnification formula can be rearranged to find the actual size or the image size:

actual size = image size ÷ magnification

image size = magnification × actual size

• Examiner reports show that students lose marks by:

  • not converting the lengths to the same unit before dividing

  • measuring in centimetres when the question asks for millimetres

  • not rounding the answer to the number of significant figures asked for

Calculating mean, range & percentage change

• In investigations, it is common to repeat a measurement several times; this makes it possible to check for anomalous results

• It is also possible to use a set of results to calculate values such as:

  • mean

  • range

  • percentage change

Mean

• The mean gives a single value that represents the whole set of results

• Calculating a mean of repeat results reduces the effect of random errors, so it is a better estimate than any single result

• To calculate the mean, add up all the values and divide by the number of values:

mean = total of the values ÷ number of values

• Leave out any anomalous results before calculating the mean

• If the question tells you which values to use, use only those

• Give your calculated mean value to the same precision as the measured data points

Range

• The range shows how spread out a set of results is

• To calculate the range, subtract the lowest value from the highest value:

range = highest value − lowest value

Percentage change

• The percentage change shows how much a value has changed compared with where it started

• It lets you compare changes fairly when the starting values are different, e.g. potato cylinders with different starting masses

• To calculate a percentage change, divide the change by the original value and multiply by 100:

percentage change = (change ÷ original value) × 100

• If a value has gone down, show this with a minus sign, or say that it is a percentage decrease

• Examiner reports show that students lose marks by:

  • including values in a mean that the question did not ask them to use

  • rounding values incorrectly

  • dividing by the final value, not the original value, in a percentage change

Rates, gradients & surface area to volume ratio

Rates

• A rate is how much something changes in a set period of time

  • E.g. the volume of oxygen produced per minute during photosynthesis

• To calculate a rate, divide the change by the time taken:

rate = change ÷ time

• The unit of a rate includes a quantity and time

  • E.g. for rate of photosynthesis the unit might be cm³ per min:

    • quantity = volume of oxygen (cm³)

    • time = per minute (min)

• Some investigations measure the time taken for a fixed change to happen, e.g. the time for a colour change; a shorter time means a faster rate

  • If the change is the same each time, you can compare rates using 1 ÷ time

  • The unit is per second or per minute

Gradients and rate

• On a graph with time on the x-axis, the gradient of the line is the rate at which the quantity on the y-axis is changing

  • A steeper line means a faster rate

• For a straight line, calculate the gradient using a large triangle drawn on the line:

gradient = change in y ÷ change in x

• The unit of the gradient is taken from the unit of y and the unit of x

  • E.g. if y is volume in cm³ and x is time in min, then the unit is cm³ per min

Surface area to volume ratio

• A ratio compares two quantities

  • For surface area to volume ratio, write the surface area first, then the volume

surface area : volume

• To calculate the surface area to volume ratio of a cube:

  • surface area = area of one face × 6 = l x l x 6

    • Where 6 = number of faces and l = length of one side

  • volume = l × l × l

• For a cuboid, the faces are not all the same, so work out the area of all six faces (there are 3 pairs of matching faces) and add them together

• To make ratios easy to compare, divide both numbers by the volume so that the ratio is written as … : 1

  • E.g. a cube with sides of 2 cm has a surface area of 6 × 2 × 2 = 24 cm² and a volume of 2 × 2 × 2 = 8 cm³

  • The ratio is 24 : 8, which simplifies to 3 : 1 (24 ÷ 8 = 3 and 8 ÷ 8 = 1)

Unit conversions, significant figures & decimal places

Unit conversions

• Calculations often involve sets of numbers that have different units; in order to determine the correct answer you need to convert units so that they match

  • Convert values to the same unit before you calculate, not afterwards

    • This reduces the risk of forgetting to convert one of the values

• Common conversions include:

Significant figures and decimal places

• Significant figures and decimal places are not the same thing, so check which one the question asks for

  • Significant figures count from the first digit that is not zero

    • E.g. 0.0487 to two significant figures is 0.049

    • Zeros after the first digit do count, e.g. 0.0450 has three significant figures

  • Decimal places count the digits after the decimal point

    • E.g. 0.0487 to two decimal places is 0.05

• Give a value to no more decimal places than the scale you read it from allows

  • When you measure, record the value to the precision of the instrument

    • E.g. a ruler marked in millimetres can only be read to the nearest millimetre

  • When you calculate, do not give more figures than your data support, even if your calculator shows many digits

    • E.g. if the lengths 12 mm, 13 mm and 13 mm give a mean of 12.6666..., give 13 mm (the same precision as the measured instrument)

• Examiner reports show that students lose marks by:

  • confusing significant figures with decimal places

  • not rounding to the number of significant figures or decimal places asked for

Checking your answer

• Before you move on from calculation questions in exams, it is a good idea to carry out some checks, for example:

  • size

    • Think about the sizes of different organisms, or parts of organisms, in relation to each other, e.g. a cell should be far smaller than a leaf

    • The magnification for an enlarged image should be greater than ×1, e.g. an answer of ×0.15 suggests that a unit conversion has been missed

    • If an answer looks far too big or far too small, check first that you converted every value into the same unit

  • direction

    • If something decreased, the percentage change should be negative, or described as a decrease

  • percentages

    • A percentage change can be over 100 %, e.g. a mass that more than doubles

    • A percentage of a whole group, e.g. the percentage of people who have been vaccinated, cannot be over 100 %

  • units

    • Units should be included, unless the answer is a magnification or a ratio

    • Check that the units are correct, e.g. don't mix up cm² and cm³, and use min for minutes, not m (m is the symbol for metres)

  • rounding

    • Ensure that the answer has been rounded correctly, and given to the correct number of decimal places or significant figures

• Show your working, so you can gain marks for correct steps even if the final answer is wrong