Chapter 1: Apply basic arithmetic operation

Applied Biology professionals in Kenya frequently encounter quantitative data that require precise calculations to analyze biological measurements, laboratory results, and experimental data. Mastery of basic arithmetic operations such as addition and subtraction is essential for accurate data interpretation and decision-making in fields like agriculture, environmental biology, and medical laboratory science. This chapter focuses on applying addition and subtraction to different types of numbers: natural numbers, integers, fractions, and decimals, all within contexts relevant to applied biology.

1.1 Addition and subtraction

Addition combines quantities and subtraction finds what is left or how much one quantity differs from another. In applied biology these operations are used every time counts, volumes and masses are totalled or compared: the colonies on two plates, the water left in a stock bottle, the change in temperature of an incubator or the soil collected from a plot. The rules differ slightly for each type of number, so natural numbers, integers, fractions and decimals are treated in turn.

Two rules hold for every type of number. Addition does not depend on order, so a + b = b + a, and three numbers can be added in any grouping, so (a + b) + c = a + (b + c). Subtraction does depend on order, so a - b is not the same as b - a. Every subtraction can be checked by adding the answer back: if a - b = d, then d + b must equal a.

1.1.1 Natural numbers

Natural numbers are the counting numbers 1, 2, 3, 4 and so on. They answer the question "how many?", such as the number of samples, organisms or test tubes. Zero is sometimes included, in which case the set is called the whole numbers. Adding natural numbers always gives a natural number. Subtracting a smaller natural number from a larger one gives a natural number, but subtracting a larger one from a smaller one does not, and negative numbers are needed for that case, as set out under integers.

The column method lines up digits of the same place value (units, tens, hundreds) and works from the right. In addition, when a column totals 10 or more, the tens digit is carried to the next column. In subtraction, when the top digit is smaller than the bottom digit, 1 is borrowed from the next column to the left and becomes 10 in the column being worked.

Addition: 478 + 356
  478
+ 356
-----
  834      (8 + 6 = 14, write 4 carry 1; 7 + 5 + 1 = 13, write 3 carry 1; 4 + 3 + 1 = 8)

Subtraction: 1204 - 685
  1204
-  685
------
   519     (borrowing in the units, tens and hundreds columns)

Steps for adding or subtracting natural numbers:

1. Decide from the wording whether quantities are being combined (add) or compared or removed (subtract).
2. Write the numbers under one another with the units in one column.
3. Work from the right, column by column, carrying or borrowing as needed.
4. Write the answer with its unit, such as plants, samples or tubes.
5. Check by an estimate: round each number and see that the answer is about the right size.
6. Check by the opposite operation: subtract the answer from the total, or add it back.

Worked examples

Example 1: a biologist counts 15 maize plants in one plot and 27 in another. What is the total?

15 + 27 = 42

Answer: 42 maize plants.

Example 2: a technician prepares 120 culture tubes, and 45 of them are inoculated on the first day. How many tubes are still to be inoculated?

120 - 45 = 75

Answer: 75 tubes. Check: 75 + 45 = 120.

Example 3: a field team delivers 200 water samples to the bench and 38 are discarded because of contamination. How many are left for analysis?

200 - 38 = 162

Answer: 162 samples.

Example 4: a biologist observes 18 birds in one tree and 23 in a nearby tree. How many birds are there in total?

18 + 23 = 41

Answer: 41 birds.

Example 5: a field technician records 350 soil samples at a site, and 125 are sent away for testing. How many samples remain in the field?

350 - 125 = 225

Answer: 225 samples.

1.1.2 Integers

Integers are the whole numbers and their negatives: ... -3, -2, -1, 0, 1, 2, 3 ... They are used when a quantity can fall below a reference value or when a change can be a gain or a loss: temperatures below zero in a cold room, a water level below a marker, a fall in a population or a drop in pH. On a number line, positive integers lie to the right of zero and negative integers to the left, and adding a positive number moves to the right while adding a negative number moves to the left.

The rules for adding integers:

• Two positive integers: add them, and the sum is positive.
• Two negative integers: add their sizes, and the sum is negative: (-3) + (-4) = -7.
• One positive and one negative integer: subtract the smaller size from the larger size and give the answer the sign of the integer with the larger size: 9 + (-4) = 5 and 4 + (-9) = -5.

Subtracting an integer is the same as adding its opposite: a - b = a + (-b). Subtracting a negative number therefore adds: 4 - (-6) = 4 + 6 = 10. A change in a quantity is always final value minus initial value, so a negative change means a fall.

Steps for adding or subtracting integers:

1. Write the calculation with every negative number in brackets.
2. Turn every subtraction into the addition of the opposite.
3. If the signs are the same, add the sizes and keep the sign.
4. If the signs differ, subtract the smaller size from the larger and keep the sign of the larger.
5. Write the result with its unit and say in words whether it is a rise or a fall.
6. Check on a number line.

Worked examples

Example 1: the temperature in a cold room falls from 5 °C to -3 °C overnight. What is the change in temperature?

Change = final - initial
Change = -3 - 5
Change = -3 + (-5)
Change = -8 °C

Answer: the temperature fell by 8 °C.

Example 2: a population of bacteria increases by 120 in one hour and then decreases by 75 in the next hour. What is the net change?

Net change = 120 + (-75) = 45

Answer: a net increase of 45 bacteria.

Example 3: a water-level logger in a stream records +12 cm relative to a fixed marker at the start of a storm and -5 cm after it. What is the change in level?

Change = -5 - 12 = -17 cm

Answer: the level fell by 17 cm. Reversed, from -5 cm back up to +12 cm, the change is 12 - (-5) = 12 + 5 = +17 cm.

Example 4: a sample stored at 2 °C is moved to a freezer that lowers its temperature by 7 °C. What is the final temperature?

Final temperature = 2 + (-7) = -5 °C

Answer: -5 °C.

Example 5: a technician records a pH change of -2 units followed by a change of +5 units. What is the overall change?

Overall change = -2 + 5 = 3

Answer: an overall increase of 3 pH units.

Example 6: find the difference between a thermometer reading of 4 °C and a reading of -6 °C.

Difference = 4 - (-6) = 4 + 6 = 10 °C

Answer: 10 °C.

1.1.3 Fractions

A fraction a/b describes a part of a whole: a is the numerator, the number of parts taken, and b is the denominator, the number of equal parts in the whole. Fractions describe proportions of species in a sample, the fraction of a solution used, the share of clay in a soil or the part of a plate that is covered by growth. Fractions can only be added or subtracted when they have the same denominator, because only parts of the same size can be counted together.

If the denominators are the same, add or subtract the numerators and keep the denominator. If the denominators differ, first rewrite both fractions with a common denominator, which is any common multiple of the denominators. The lowest common multiple gives the smallest numbers. A fraction is rewritten without changing its value by multiplying the numerator and the denominator by the same number.

General rule
a/c + b/d = (ad + bc) / cd
a/c - b/d = (ad - bc) / cd

Steps for adding or subtracting fractions:

1. Find the lowest common multiple of the denominators.
2. Rewrite each fraction with that denominator by multiplying its numerator and denominator by the same number.
3. Add or subtract the numerators and keep the common denominator.
4. Simplify by dividing numerator and denominator by their largest common factor.
5. If the answer is greater than 1, write it as a mixed number, such as 9/8 = 1 1/8.
6. For mixed numbers, convert to improper fractions first, or add the whole numbers and the fractions separately.

Worked examples

Example 1: a flask holds 3/4 litre of plant extract and another 1/4 litre is added. What is the total volume?

3/4 + 1/4 = (3 + 1)/4 = 4/4 = 1

Answer: 1 litre. The denominators are equal, so only the numerators are added.

Example 2: a technician has 5/6 kg of fertiliser and uses 1/3 kg. How much remains?

1/3 = 2/6
5/6 - 2/6 = 3/6 = 1/2

Answer: 1/2 kg remains.

Example 3: a soil sample contains 2/5 clay and 3/10 silt. What fraction of the sample is clay and silt together?

2/5 = 4/10
4/10 + 3/10 = 7/10

Answer: 7/10 of the sample.

Example 4: 7/8 of a hectare of a trial site is planted with maize and 1/4 of a hectare with beans. What area is planted in total?

1/4 = 2/8
7/8 + 2/8 = 9/8 = 1 1/8

Answer: 1 1/8 hectares.

Example 5: a petri dish is 3/4 covered with bacterial growth, and a swab then removes growth from an area equal to 2/5 of the dish, all of it inside the grown area. What fraction of the dish is still covered?

Common denominator 20: 3/4 = 15/20 and 2/5 = 8/20
15/20 - 8/20 = 7/20

Answer: 7/20 of the dish is still covered.

Example 6: subtract 1 3/4 litres from 2 1/2 litres of buffer.

2 1/2 = 5/2 = 10/4 and 1 3/4 = 7/4
10/4 - 7/4 = 3/4

Answer: 3/4 litre.

1.1.4 Decimals

Decimals write parts of a whole using place value to the right of the decimal point: tenths, hundredths, thousandths and so on. They are the form in which most laboratory instruments report results, such as the mass of a reagent on a balance, a pipetted volume or the absorbance of a sample. Adding and subtracting decimals follows the same rules as whole numbers, provided that digits of the same place value are placed in the same column.

The decimal points must be lined up under one another. Where a number has fewer decimal places than another, zeros are added at the end so that every number has the same number of places. Adding zeros after the last decimal digit does not change the value, so 2.5 = 2.50. The decimal point in the answer sits directly below the points in the numbers.

Steps for adding or subtracting decimals:

1. Write the numbers under one another with the decimal points in one vertical line.
2. Add zeros so that every number has the same number of decimal places.
3. Add or subtract as for whole numbers, starting at the right, carrying or borrowing as needed.
4. Place the decimal point in the answer directly below the others.
5. Write the unit and keep the number of decimal places that the instrument allows.
6. Check by estimating with whole numbers: 3.75 + 2.5 is about 4 + 3 = 7, and the answer 6.25 is near 7.

Worked examples

Example 1: a technician measures 3.75 mL of reagent and adds 2.5 mL more. What is the total volume?

  3.75
+ 2.50
------
  6.25

Answer: 6.25 mL.

Example 2: 12.6 litres of pesticide solution are sprayed on a field plot, but 3.45 litres leak from the sprayer. How much reaches the plants?

  12.60
-  3.45
-------
   9.15

Answer: 9.15 litres.

Example 3: a soil sample weighs 25.38 kg and 7.12 kg is removed for testing. What mass remains?

25.38 - 7.12 = 18.26

Answer: 18.26 kg.

Example 4: a water tank holds 45.75 litres and 13.8 litres is added. What is the new volume?

  45.75
+ 13.80
-------
  59.55

Answer: 59.55 litres.

Example 5: a researcher records 0.256 g of enzyme in one sample and 0.874 g in another. What is the combined mass?

  0.256
+ 0.874
-------
  1.130

Answer: 1.130 g.

Chapter Summary

This chapter covered the fundamental arithmetic operations of addition and subtraction applied to natural numbers, integers, fractions, and decimals, highlighting how each number type requires specific handling. It then progressed to multiplication and division, demonstrating their application across the same sets of numbers with clear computational methods. The chapter introduced rational and irrational numbers, explaining their differences and significance in numerical calculations. Finally, it explored ratios, proportions, and percentages, detailing both direct and inverse proportions and their practical uses in problem-solving. Through these topics, learners developed a comprehensive understanding of basic arithmetic operations and their relevance to various types of numbers and mathematical relationships. This foundation is essential for tackling more complex mathematical challenges in science and engineering contexts.