By the end of this chapter, you will be able to:
These skills are essential because they build a strong foundation in math that you will use every day in your trade to solve real-world problems efficiently and confidently.
Compiled by: Kennedy Wachira
Apply basic arithmetic operation is fundamental for Science Laboratory Technology professionals in Kenya, as precise quantitative calculations underpin accurate experimental results and data analysis. Mastery of addition and subtraction across different number types ensures laboratory technicians can handle measurements, chemical quantities, and data recording reliably. This chapter focuses on performing addition and subtraction with natural numbers, integers, fractions, and decimals, emphasizing practical applications in laboratory contexts.
Natural numbers are positive whole numbers starting from 1 upwards, commonly used in counting and measuring discrete quantities in laboratories such as sample counts or reagent volumes. Addition and subtraction of natural numbers form the basis for handling simple quantitative data and inventory calculations in laboratory settings.
The fundamental operations are:
$$ \text{Addition: } S = a + b $$
$$ \text{Subtraction: } D = a, b \quad \text{where } a \geq b $$
Example 1: A laboratory technician counts 35 test tubes in one batch and 47 in another. Find the total number of test tubes.
Given: \(a = 35\), \(b = 47\)
$$ S = a + b $$
$$ S = 35 + 47 $$
$$ S = 82 $$
Answer: 82 test tubes
Example 2: A sample container initially has 120 ml of solution. After using 45 ml for analysis, how much remains?
Given: \(a = 120\) ml, \(b = 45\) ml
$$ D = a, b $$
$$ D = 120 - 45 $$
$$ D = 75 \text{ ml} $$
Answer: 75 ml remaining
Example 3: A laboratory orders 150 pipette tips. After distributing 78 tips to different sections, how many tips are left?
Given: \(a = 150\), \(b = 78\)
$$ D = a, b $$
$$ D = 150 - 78 $$
$$ D = 72 $$
Answer: 72 pipette tips left
Example 4: Two batches of reagents contain 95 and 130 bottles respectively. Find the total number of bottles.
Given: \(a = 95\), \(b = 130\)
$$ S = a + b $$
$$ S = 95 + 130 $$
$$ S = 225 $$
Answer: 225 bottles
Example 5: A laboratory receives 400 ml of a chemical but uses 260 ml for preparation. Determine the volume remaining.
Given: \(a = 400\) ml, \(b = 260\) ml
$$ D = a, b $$
$$ D = 400 - 260 $$
$$ D = 140 \text{ ml} $$
Answer: 140 ml remaining
A laboratory has 125 test tubes in one cabinet and 230 in another. Calculate the total number of test tubes. (3 marks)
If a sample weighs 560 g and 275 g is used for testing, find the remaining weight. (3 marks)
A chemical stock has 800 ml. After dispensing 345 ml, how much is left? (3 marks)
Two reagent bottles contain 120 ml and 85 ml respectively. Find their combined volume. (3 marks)
A laboratory receives 500 units of consumables and distributes 175 units. Determine the remaining units. (3 marks)
Integers extend natural numbers to include zero and negative numbers, which are essential for laboratory calculations involving temperature changes, pH shifts, or deviations from reference values. Adding and subtracting integers allows technicians to interpret measurements that may fall below zero or represent increases and decreases in quantities.
The operations follow these rules:
$$ \text{Addition: } S = a + b $$
$$ \text{Subtraction: } D = a, b = a + (-b) $$
Example 1: The temperature in a cold storage changes from -5°C to 3°C. Calculate the temperature increase.
Given: Initial \(a = -5^\circ C\), Final \(b = 3^\circ C\)
$$ \text{Change} = b, a $$
$$ \text{Change} = 3 - (-5) $$
$$ \text{Change} = 3 + 5 $$
$$ \text{Change} = 8^\circ C $$
Answer: 8°C increase
Example 2: A pH meter reading changes from 7 to 4. Find the change in pH.
Given: Initial \(a = 7\), Final \(b = 4\)
$$ \text{Change} = b, a $$
$$ \text{Change} = 4 - 7 $$
$$ \text{Change} = -3 $$
Answer: pH decreased by 3 units
Example 3: A sample temperature drops from 10°C to -2°C. Calculate the temperature change.
Given: Initial \(a = 10^\circ C\), Final \(b = -2^\circ C\)
$$ \text{Change} = b, a $$
$$ \text{Change} = -2 - 10 $$
$$ \text{Change} = -12^\circ C $$
Answer: 12°C decrease
Example 4: A laboratory's pH level changes from -1 to 2 (hypothetical scale for illustration). Find the net change.
Given: Initial \(a = -1\), Final \(b = 2\)
$$ \text{Change} = b, a $$
$$ \text{Change} = 2 - (-1) $$
$$ \text{Change} = 2 + 1 $$
$$ \text{Change} = 3 $$
Answer: Increase of 3 units
Example 5: The reading on a scale shifts from -8 to -3. Calculate the change.
Given: Initial \(a = -8\), Final \(b = -3\)
$$ \text{Change} = b, a $$
$$ \text{Change} = -3 - (-8) $$
$$ \text{Change} = -3 + 8 $$
$$ \text{Change} = 5 $$
Answer: Increase of 5 units
A freezer temperature changes from -10°C to 0°C. Calculate the temperature increase. (3 marks)
The pH of a solution changes from 6 to 2. Find the pH change. (3 marks)
A sample’s temperature drops from 12°C to -4°C. Determine the temperature change. (3 marks)
A meter reading changes from -5 to 7. Calculate the net change. (3 marks)
The temperature changes from -15°C to -10°C. Find the temperature difference. (3 marks)
Fractions represent parts of a whole and are vital in laboratory calculations such as reagent dilutions, concentration ratios, and measurements expressed as parts of units. Adding and subtracting fractions require a common denominator to combine quantities accurately.
The operations are:
Addition: Convert fractions to a common denominator then add numerators.
Subtraction: Convert fractions to a common denominator then subtract numerators.
$$ \text{Addition: } \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} $$
$$ \text{Subtraction: } \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} $$
Example 1: Add \( \frac{1}{4} \) litre and \( \frac{1}{2} \) litre of reagent.
Given: \( \frac{a}{b} = \frac{1}{4} \), \( \frac{c}{d} = \frac{1}{2} \)
$$ \frac{1}{4} + \frac{1}{2} = \frac{1 \times 2 + 1 \times 4}{4 \times 2} $$
$$ = \frac{2 + 4}{8} $$
$$ = \frac{6}{8} = \frac{3}{4} \text{ litre} $$
Answer: \(\frac{3}{4}\) litre
Example 2: Subtract \( \frac{2}{5} \) ml from \( \frac{3}{4} \) ml volume.
Given: \( \frac{a}{b} = \frac{3}{4} \), \( \frac{c}{d} = \frac{2}{5} \)
$$ \frac{3}{4} - \frac{2}{5} = \frac{3 \times 5 - 2 \times 4}{4 \times 5} $$
$$ = \frac{15 - 8}{20} $$
$$ = \frac{7}{20} \text{ ml} $$
Answer: \(\frac{7}{20}\) ml
Example 3: Add \( \frac{5}{6} \) grams and \( \frac{1}{3} \) grams of a chemical.
Given: \( \frac{a}{b} = \frac{5}{6} \), \( \frac{c}{d} = \frac{1}{3} \)
$$ \frac{5}{6} + \frac{1}{3} = \frac{5 \times 3 + 1 \times 6}{6 \times 3} $$
$$ = \frac{15 + 6}{18} $$
$$ = \frac{21}{18} = \frac{7}{6} \text{ grams} $$
Answer: \(\frac{7}{6}\) grams
Example 4: Subtract \( \frac{7}{8} \) litres from \( \frac{3}{2} \) litres of solution.
Given: \( \frac{a}{b} = \frac{3}{2} \), \( \frac{c}{d} = \frac{7}{8} \)
$$ \frac{3}{2} - \frac{7}{8} = \frac{3 \times 8 - 7 \times 2}{2 \times 8} $$
$$ = \frac{24 - 14}{16} $$
$$ = \frac{10}{16} = \frac{5}{8} \text{ litres} $$
Answer: \(\frac{5}{8}\) litres
Example 5: Add \( \frac{4}{9} \) litres and \( \frac{2}{3} \) litres of liquid.
Given: \( \frac{a}{b} = \frac{4}{9} \), \( \frac{c}{d} = \frac{2}{3} \)
$$ \frac{4}{9} + \frac{2}{3} = \frac{4 \times 3 + 2 \times 9}{9 \times 3} $$
$$ = \frac{12 + 18}{27} $$
$$ = \frac{30}{27} = \frac{10}{9} \text{ litres} $$
Answer: \(\frac{10}{9}\) litres
Add \( \frac{2}{3} \) litre and \( \frac{3}{4} \) litre of chemical solution. (4 marks)
Subtract \( \frac{5}{6} \) ml from \( \frac{7}{8} \) ml of reagent. (4 marks)
Add \( \frac{7}{10} \) grams and \( \frac{1}{5} \) grams of compound. (4 marks)
Subtract \( \frac{3}{7} \) litre from \( \frac{5}{3} \) litres of liquid. (4 marks)
Add \( \frac{9}{12} \) litres and \( \frac{2}{5} \) litres of solution. (4 marks)
Decimals are essential in laboratory measurements for precision in volumes, weights, concentrations, and time intervals. Adding and subtracting decimals requires aligning decimal points to ensure accurate calculation of quantities.
The operations are:
Addition: Align decimal points, add digits column-wise.
Subtraction: Align decimal points, subtract digits column-wise.
Example 1: Add 12.5 ml and 3.75 ml of reagent.
Given: \(a = 12.5\) ml, \(b = 3.75\) ml
$$ S = a + b $$
$$ S = 12.5 + 3.75 $$
$$ S = 16.25 \text{ ml} $$
Answer: 16.25 ml
Example 2: Subtract 4.3 g from 9.75 g of sample.
Given: \(a = 9.75\) g, \(b = 4.3\) g
$$ D = a, b $$
$$ D = 9.75 - 4.3 $$
$$ D = 5.45 \text{ g} $$
Answer: 5.45 g
Example 3: Add 0.125 litres and 1.875 litres of solution.
Given: \(a = 0.125\) litres, \(b = 1.875\) litres
$$ S = a + b $$
$$ S = 0.125 + 1.875 $$
$$ S = 2.0 \text{ litres} $$
Answer: 2.0 litres
Example 4: Subtract 0.65 ml from 3.2 ml of liquid.
Given: \(a = 3.2\) ml, \(b = 0.65\) ml
$$ D = a, b $$
$$ D = 3.2 - 0.65 $$
$$ D = 2.55 \text{ ml} $$
Answer: 2.55 ml
Example 5: Add 5.5 g, 3.25 g, and 1.125 g of chemical samples.
Given: \(a = 5.5\) g, \(b = 3.25\) g, \(c = 1.125\) g
$$ S = a + b + c $$
$$ S = 5.5 + 3.25 + 1.125 $$
$$ S = 8.75 + 1.125 $$
$$ S = 9.875 \text{ g} $$
Answer: 9.875 g
Add 14.75 ml and 6.5 ml of solution. (3 marks)
Subtract 2.8 g from 7.45 g of sample. (3 marks)
Add 0.375 litres and 0.625 litres of reagent. (3 marks)
Subtract 1.25 ml from 5.5 ml of liquid. (3 marks)
Add 10.5 g, 2.25 g, and 3.125 g of chemical substances. (4 marks)
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Create a free accountThis chapter covered the application of basic arithmetic operations including addition and subtraction performed on natural numbers, integers, fractions, and decimals. It then explored multiplication and division applied to the same sets of numbers, emphasizing the rules and methods specific to each type. The chapter further examined ratios, proportions, and percentages, distinguishing between direct and inverse proportion and their practical implications. A detailed study of indices followed, starting with the concept of bases, then progressing through the fundamental laws governing indices. Finally, the chapter addressed indicial equations, demonstrating how to solve problems involving unknown powers. Together, these topics establish a strong foundation in arithmetic operations essential for further study in science and engineering fields.
A laboratory technician measures two volumes of chemical solutions: 125 ml and 378 ml. Calculate the total volume when these are mixed. (2 marks)
In an experiment, a solution’s temperature dropped from 25°C to -8°C. Find the total change in temperature. (2 marks)
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