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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.
Natural numbers are the counting numbers starting from 1 upwards (1, 2, 3, …). In applied biology, natural numbers often represent counts such as the number of samples, organisms, or test tubes. Addition combines these counts, while subtraction finds the difference between amounts.
The basic formulas are:
Addition:$$ S = a + b $$
where \(a\) and \(b\) are natural numbers and \(S\) is their sum.
Subtraction:$$ D = a, b $$ where \(a\) and \(b\) are natural numbers and \(D\) is the difference, with \(a \geq b\).
Example 1: A biologist counts 15 maize plants in one plot and 27 in another. What is the total number of maize plants counted?
Given: \(a=15\) plants, \(b=27\) plants$$ S = a + b $$$$ S = 15 + 27 $$$$ S = 42 $$
Answer: 42 maize plants
Example 2: A farmer has 120 eggs collected from a poultry farm. After selling 45 eggs, how many eggs remain?
Given: \(a=120\) eggs, \(b=45\) eggs$$ D = a, b $$$$ D = 120 - 45 $$$$ D = 75 $$
Answer: 75 eggs remain
Example 3: In a laboratory, 200 blood samples were collected. 38 samples were discarded due to contamination. How many samples are left for analysis?
Given: \(a=200\) samples, \(b=38\) samples$$ D = a, b $$$$ D = 200 - 38 $$$$ D = 162 $$
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?
Given: \(a=18\), \(b=23\)$$ S = a + b $$$$ S = 18 + 23 $$$$ S = 41 $$
Answer: 41 birds
Example 5: A field technician recorded 350 soil samples from a farm. If 125 samples are taken to a lab for testing, how many samples remain in the field?
Given: \(a=350\), \(b=125\)$$ D = a, b $$$$ D = 350 - 125 $$$$ D = 225 $$
Answer: 225 samples
Integers include positive and negative whole numbers including zero (… -3, -2, -1, 0, 1, 2, 3 …). In applied biology, integers may represent temperature changes, pH level deviations, or population changes (increases or decreases). Addition and subtraction of integers follow specific rules depending on the sign.
Addition:
Subtraction:
Subtracting an integer is equivalent to adding its opposite:$$ a - b = a + (-b) $$
Example 1: The temperature in a laboratory dropped from 5°C to -3°C overnight. What was the change in temperature?
Given: Initial temperature \(a=5^\circ C\), final temperature \(b=-3^\circ C\)
Change in temperature:$$ \Delta T = b, a $$$$ \Delta T = -3 - 5 $$$$ \Delta T = -8^\circ C $$
Answer: Temperature dropped by 8°C
Example 2: A population of bacteria increased by 120 in one hour and then decreased by 75 in the next hour. What is the net change?
Given: Increase \(a=+120\), decrease \(b=-75\)$$ \text{Net change} = a + b $$$$ \text{Net change} = 120 + (-75) $$$$ \text{Net change} = 45 $$
Answer: Net increase of 45 bacteria
Example 3: The pH level of a solution changed from -1 to 4 units. What is the difference in pH level?
Given: initial pH \(a=-1\), final pH \(b=4\)$$ D = b, a $$$$ D = 4 - (-1) $$$$ D = 4 + 1 $$$$ D = 5 $$
Answer: Increase of 5 pH units
Example 4: A farmer’s fish pond water temperature was 2°C in the morning and dropped by 7°C at night. What is the final temperature?
Given: Initial \(a=2^\circ C\), change \(b=-7^\circ C\)$$ \text{Final temperature} = a + b $$$$ \text{Final temperature} = 2 + (-7) $$$$ \text{Final temperature} = -5^\circ C $$
Answer: Final temperature is -5°C
Example 5: A lab technician recorded a pH change of -2 units followed by an increase of 5 units. What is the overall pH change?
Given: \(a=-2\), \(b=+5\)$$ \text{Overall change} = a + b $$$$ \text{Overall change} = -2 + 5 $$$$ \text{Overall change} = 3 $$
Answer: Overall increase of 3 pH units
Fractions represent parts of a whole and are common in biological measurements such as proportions of species, concentration ratios, or enzyme activity fractions. Addition and subtraction require a common denominator.
General formula:
For fractions \(\frac{a}{c}\) and \(\frac{b}{d}\),
Addition:$$ \frac{a}{c} + \frac{b}{d} = \frac{ad + bc}{cd} $$
Subtraction:$$ \frac{a}{c} - \frac{b}{d} = \frac{ad - bc}{cd} $$
Example 1: A solution contains \(\frac{3}{4}\) litre of extract. Another \(\frac{1}{4}\) litre is added. What is the total volume?
Given: \(\frac{3}{4}\) litre, \(\frac{1}{4}\) litre$$ \frac{3}{4} + \frac{1}{4} = \frac{3 \times 1 + 1 \times 4}{4 \times 4} \text{ (not needed as denominators are same)} $$
Since denominators are equal:$$ \frac{3}{4} + \frac{1}{4} = \frac{3+1}{4} = \frac{4}{4} = 1 $$
Answer: 1 litre
Example 2: A biologist has \(\frac{5}{6}\) kg of fertilizer and uses \(\frac{1}{3}\) kg. How much fertilizer remains?
Given: \(\frac{5}{6}\) kg, \(\frac{1}{3}\) kg
Find common denominator 6: \(\frac{1}{3} = \frac{2}{6}\)$$ \frac{5}{6} - \frac{2}{6} = \frac{5 - 2}{6} = \frac{3}{6} = \frac{1}{2} $$
Answer: \(\frac{1}{2}\) kg fertilizer remains
Example 3: A soil sample contains \(\frac{2}{5}\) clay and \(\frac{3}{10}\) silt. What is the total fraction of clay and silt?
Given: \(\frac{2}{5}\), \(\frac{3}{10}\)
Convert \(\frac{2}{5} = \frac{4}{10}\)$$ \frac{4}{10} + \frac{3}{10} = \frac{7}{10} $$
Answer: \(\frac{7}{10}\) of the sample
Example 4: A farmer allocates \(\frac{7}{8}\) of a hectare for maize and \(\frac{1}{4}\) for beans. What is the total area allocated?
Given: \(\frac{7}{8}\), \(\frac{1}{4}\)
Convert \(\frac{1}{4} = \frac{2}{8}\)$$ \frac{7}{8} + \frac{2}{8} = \frac{9}{8} = 1 \frac{1}{8} $$ hectares
Answer: 1 \(\frac{1}{8}\) hectares
Example 5: A petri dish is \(\frac{3}{4}\) full of bacterial culture. After removing \(\frac{2}{5}\), what fraction remains?
Given: \(\frac{3}{4}\), \(\frac{2}{5}\)
Find common denominator 20:
\(\frac{3}{4} = \frac{15}{20}\), \(\frac{2}{5} = \frac{8}{20}\)$$ \frac{15}{20} - \frac{8}{20} = \frac{7}{20} $$
Answer: \(\frac{7}{20}\) of the culture remains
Decimals represent fractional parts and are used extensively in applied biology for measurements such as weights, volumes, concentrations, and distances. Addition and subtraction of decimals follow the same principles as whole numbers, aligning decimal points.
Example 1: A lab technician measures 3.75 ml of reagent and adds 2.5 ml more. What is the total volume?
Given: 3.75 ml, 2.5 ml$$ S = 3.75 + 2.5 $$$$ S = 3.75 + 2.50 $$$$ S = 6.25 $$
Answer: 6.25 ml
Example 2: A farmer sprayed 12.6 litres of pesticide but 3.45 litres leaked. How much pesticide was effectively used?
Given: 12.6 litres, 3.45 litres$$ D = 12.6 - 3.45 $$
Align decimals:$$ D = 12.60 - 3.45 $$$$ D = 9.15 $$
Answer: 9.15 litres used
Example 3: A soil sample weighs 25.38 kg. After removing 7.12 kg for testing, what is the remaining weight?
Given: 25.38 kg, 7.12 kg$$ D = 25.38 - 7.12 $$$$ D = 18.26 $$
Answer: 18.26 kg
Example 4: A water tank contains 45.75 litres. An additional 13.8 litres is added. What is the new volume?
Given: 45.75 litres, 13.8 litres$$ S = 45.75 + 13.8 $$$$ S = 45.75 + 13.80 $$$$ S = 59.55 $$
Answer: 59.55 litres
Example 5: A researcher recorded 0.256 grams of enzyme in one sample and 0.874 grams in another. What is their combined mass?
Given: 0.256 g, 0.874 g$$ S = 0.256 + 0.874 $$$$ S = 1.130 $$
Answer: 1.130 grams
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Create a free accountThis 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.
A laboratory technician counts 245 bacteria colonies on one petri dish and 378 colonies on another. What is the total number of colonies? (2 marks)
A biologist measured the pH values of two soil samples as -3 and +7. Calculate the difference between these pH values. (2 marks)
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