Proportional Reasoning

This section covers the concepts of ratio, rate and proportion as methods for comparing quantities. A ratio compares two quantities of the same type, while a rate compares quantities of different types, for example, speed or price per litre of fuel. Proportion then shows the equivalence of two ratios or rates, highlighting how relationships scale in practical situations. Through these concepts you will gain essential skills for real-world problem-solving and quantitative analysis.

KEY IDEAS

• Distance-time graphs represent movement; their slope indicates speed and flat sections indicate rest.

• Proportion: Proportion is an equation that shows that two ratios or rates are related by a constant.

• Rate: A rate compares two quantities of different types, such as distance over time (e.g., miles per hour) or price per unit.

• Ratio: A ratio compares two quantities of the same type, indicating how much of one quantity there is relative to the other.

• Scaling Ratios: Ratios can be scaled up or down to create equivalent ratios, helping to set up and solve proportions in various scenarios.

• Speed is calculated as the rate of change of distance over time

• Unit Rate: The unit rate is the rate for one unit of a given quantity, for

example, kilometres per hour or cost per item.

Ratio versus Rate You have encountered the concepts of ratios and rates in Junior High School Mathematics. Now, we will explore how these concepts apply to our everyday lives.

Ratios A ratio shows how many times one number contains another. For example, if there are 10 boys and 30 girls in a classroom, the ratio of boys to girls is expressed as 10 : 30 = 1 : 3 (read as 1 to 3), meaning for every boy, there are three girls. The two main types of ratios we will discuss are part-to-part and part-to-whole ratios.

Part-to-Part Ratio

A part-to-part ratio compares the amounts of two distinct components within a mixture.

For example, consider the preparation of a popular Ghanaian dish like jollof rice.

If you are making jollof rice and you use 2 cups of rice and 1 cup of tomato paste, the ratio of rice to tomato paste is a part-to-part ratio. This means you are comparing the amount of rice to the amount of tomato paste directly, which can be expressed as 2:1. This ratio indicates that for every 2 cups of rice, you need 1 cup of tomato paste.

Part-to-Whole Ratio

On the other hand, a part-to-whole ratio compares one component to the total amount of all components in a mixture. Let’s take another example involving a popular Ghanaian beverage, sobolo (hibiscus tea).

For example, when preparing sobolo, if you use 3 cups of dried hibiscus flowers, 1 cup of sugar, and 4 cups of water, the total amount of the mixture is 3 + 1 + 4 = 8 cups. The ratio of dried hibiscus flowers to the total mixture is a part-to-whole ratio. In this case, it can be expressed as 3:8. This means that out of the total 8 cups of sobolo, 3 cups are made up of dried hibiscus flowers.

To establish a ratio, we compare two quantities with the same unit of measure.

The ratio can be written as a:b or a/b and is read as “a to b.”

We can create a ratio that compares the number of males to the number of females in your class, school, town, district, region or country. You can express it as:

Number of males : Number of females

Example 5.1

In a class containing 25 boys and 15 girls, find the ratio of:

1. Boys to girls

2. Girls to boys

Solution

1. Ratio of boys to girls = 25: 15 = 5:3

2. Ratio of girls to boys15: 25 = 3:5 Now undertake the following activities to deepen your understanding of ratios.

Activity 5.1: Understanding part-to-part ratio by mixing part of gari and part of sugar Materials:

• Gari

• Sugar

• Measuring cups

• A bowl or container Procedure:

Step 1: Measure the gari: Use a measuring cup to measure out 2 cups of gari.

Step 2: Measure the sugar: Use a measuring cup to measure out 1 cup of sugar.

Step 3: Record the measurements: Write down the measurements of gari and sugar in your notebook.

Step 4: Understand the part-to-part ratio: Recognise that the ratio of gari to sugar is 2:1. This means that for every 2 parts of gari, there is 1 part of sugar.

Step 5: Mix the gari and sugar in a bowl or container.

Step 6: Observe and record: Observe the mixture and record your findings.

Step 7: Simplify the ratio (if necessary): If the ratio is not in its simplest form, simplify it by dividing both numbers by their greatest common divisor.

Share your findings with the class, including the measurements, ratio and observations.

How does the ratio of gari to sugar affect the taste and texture of the mixture?

Activity 5.2: Finding the ratio of teaching staff to non-teaching staff and non-teaching staff to teaching staff in your school Materials:

Pen/pencil, Paper/notebook and access to school administration office.

Procedure:

Step 1: Visit the school administration office: Politely ask for the number of teaching and non-teaching staff in the school.

Step 2: Record the numbers: Write down the numbers provided by the administration.

Step 3: Find the ratio of teaching staff to non-teaching staff: For example, if there are 50 teachers and 30 non-teachers, the ratio is 50 : 30.

Step 4: Find the ratio of non-teaching staff to teaching staff: From the example above, this would be 30 : 50.

Step 5: Simplify the ratios (if necessary): If the ratios are not in their simplest form, divide both numbers by their highest common denominator. In our

example, this would make 5:3 and 3:5 respectively.

Step 6: Record and present your findings Share your ratios with classmates and compare results.

Discuss the meaning of the ratios and how they relate to school operations and planning.

Activity 5.3: Measure the height of your desk and chair and find the ratio of desk height to chair height Materials:

Ruler or measuring tape, desk, chair, pencil or pen, paper Procedure:

Step 1: Measure desk height from floor to top surface.

Step 2: Measure chair height from floor to top of seat.

Step 3: Record measurements.

Step 4: Find the ratio of desk height to chair height.

Step 5: Simplify the ratio if necessary.

Step 6: Share findings with a partner or the class.

Example 5.2

In your school football team, there are 11 players and 4 substitutes. What is the ratio of:

1. Players to substitutes.

2. Substitutes to players.

Solution

a. Number of players = 11

b. Number of substitutes = 4

1. Players to substitutes

Step 1: Write the ratio format: Players : Substitutes = 11:4

Step 2: Check for simplification: The numbers 11 and 4 have no common factor other than 1.

Step 3: Final ratio: Players to substitutes = 11 : 4.

Therefore, the ratio of players to substitutes is 11 : 4.

2. Substitutes to players

Step 1: Write the ratio format: Substitutes : Players = 4:11.

Step 2: Check for simplification: The numbers 4 and 11 have no common factor other than 1.

Step 3: Final ratio: Substitutes to players = 4 : 11.

Therefore, the ratio of substitutes to players is 4 : 11.

Example 5.3

In a community of 1200 people, there are 700 adults and the rest are children.

Find the ratio of adults to children.

Solution

Given: total population= 1,200 and the population of adults = 700 This implies that the population of children = Total population – Adults population 1 200 – 7,00 = 500 children Therefore, the ratio of adults to children is 700: 500 = 7: 5

Example 5.4

A gold-buying company called Day Break in Prestea has a workforce of 120 employees. 30 of these employees are female. Find the ratio of male employees to female employees.

Solution

Step 1: Calculate the number of male employees Total employees = 120, Female employees = 30 Male employees = Total employees – Female employees = 120 – 30 = 90

Step 2: Write the ratio Male employees : Female employees = 90:30

Step 3: Simplify the ratio Find the highest common factor (HCF) of 90 and 30, which is 30.

Divide both terms of the ratio by 30:

90 ÷ 30 = 3, 30 ÷ 30 = 1 Therefore, the ratio of male employees to female employees is 3 : 1.

Rates A rate compares two quantities of different types or units. It describes how one quantity changes in relation to another, such as the speed of a vehicle and the distance it travels.

Example 5.5

A train covers a distance of 150 km in 2 hours.

Calculate its speed.

Solution

Step 1: Recall the formula for speed Speed (or rate) = Total distance traveled________________ Total time taken

Step 2: Substitute the given values into the formula:

Distance = 150 km Time = 2 hours Speed (or rate) = Total distance traveled________________ Time taken = 150km/2 hours

Step 3: Perform the division Speed = 75km/h Therefore, the speed of the train is 75 km/h

Example 5.6

A bus travels from Accra to Kumasi, covering a distance of 250 km in 5 hours.

Determine its rate.

Solution

The rate (or speed) of the car is written as Total distance traveled________________ Total time taken = 250km______ 5hours = 50km / h Undertake the activity below to deepen your understanding of rates.

Activity 5.4: Understanding rates by calculating transport fares Materials:

- A list of transport fares for different routes to your school

- Calculators

- Paper and pencils Procedure:

Step 1: Choose a route: For example, Korkorse to Nchumuruman SHS

Step 2: Find the transport fare: Look up the transport fare for the chosen route.

Step 3: Calculate the rate: Calculate the rate of the transport fare per kilometre.

For example, if the fare is GH¢5 for a 10km journey, the rate would be Gh¢5____ 10km = 0.5 which means Gh¢0.50 per kilometre travelled.

Step 4: Record the rate: Record the rate on a piece of paper.

Step 5: Compare rates: Compare the rates for different routes with your classmates and discuss any observations.

Discuss how understanding rates can help in real-life situations, such as planning a trip or budgeting for transportation.

Example 5.7

A farm has 28 animals and the ratio of cows to goats is 4:3.

How many cows and goats are there?

Solution

Method 1:

Since the total ratio is 4 + 3 = 7, we partition the 28 animals into 7 groups of 4 as follows:

Figure 5.1: 4 groups of 4 cows Figure 5.2: 3 groups of 4 goats We observe from the figure above that the four groups of cows account for a total of 16 cows, while the three groups of goats make up a total of 12 goats.

As a result, the farm has 16 cows and 12 goats. Check, 16:12 = 4:3 Method 2:

Given that the ratio of cows to goats is 4:3, we can continuously group the animals in sets of fours for cows and sets of threes for goats until all the animals are grouped. The table below illustrates this process.

Table 5.1: Ratios

Step/group Cows Goats Ratio Total

1ˢᵗ4 3 4:3 7 2ⁿᵈ8 6 8:6 = 4:3 14 3ʳᵈ12 9 12:9 = 4:3 21 4ᵗʰ16 12 16:12: = 4:3 28 We will see that on the 4th step there are 16 cows and 12 goats making a total of 28 animals.

Method 3:

Let x be the number of parts in the ratio.

Given the total number of animals to be 28, we set up the equation: 4x + 3x = 28.

Solve for x:

7x = 28, ∴ x = 4 Calculating for the number of cows, We have 4x = 4(4) = 16, Calculating for the number of goats, we have 3x = 3(4) = 12.

Therefore, there are 16 cows and 12 goats in the farm.

If you have a ratio and you want to use it in a different situation or with a different group, we can use proportions to create equivalent ratios and work out any unknown amounts in the same way. Proportions can be seen as an equation of two equivalent ratios or rates.

Let us now engage in activities involving setting up and solving proportions to apply a given ratio or rate to a new scenario.

Example 5.8

A family uses 4 cups of groundnut paste to make groundnut soup for 8 servings.

How much groundnut paste would they need for 20 servings?

Solution

Step 1: Identify the known ratio:

4 cups of groundnut paste → 8 servings.

This can be expressed as a ratio: 4 cups/8 servings

Step 2: Set up a Proportion:

We want to find the amount of groundnut paste (x ) for 20 servings:

4 cups_ 8 servings = xcups/20 servings

Step 3: Solve the Proportion:

Using cross-multiplication:

4 × 20 = 8 × x 80 = 8x Divide both sides by 8 to isolate x :

x = 80/8 x = 10 Therefore, the family would need 10 cups of groundnut paste to make soup for 20 servings.

Example 5.9

If a pot of waakye for 4 people needs 2 cups of beans, how many cups of beans are required for 10 people?

Solution

We set up the proportion as 2cups → 4 people and xcups → 10 people ⇒ 2cups/4 people = xcups/10 people Now, solve for x :

x = 2 × 10/4 = 20/4 x = 5 So, 5 cups of beans are required for 10 people.

Example 5.10

On a map of Korkorse, 1 inch represents 10 kilometres.

How many kilometres are represented by 4 inches on the map?

Solution

Step 1: Identify the scale of the map:

The scale of the map is 1 inch represents 10 kilometres.

Step 2: Identify the number of inches to be converted:

We need to find the distance represented by 4 inches on the map.

Step 3: Set up the proportion:

Since 1 inch represents 10 kilometres, we can set up a proportion to find the distance represented by 4 inches:

1 inch/10 km = 4 inches_______ × km

Step 4: Cross-multiply and solve for x 1 × × = 4 × 10 x = 40 Therefore, the distance represented by 4 inches on the map is 40 kilometres.

Example 5.11

If a pack of sachet water (30 sachets) costs Gh¢5.00, how much would it cost for 100 sachets?

Solution

Step 1: Identify the known rate:

Gh¢5.00 → 30 sachets This can be written as a rate: Gh¢5/30 sachets

Step 2: Set Up a Proportion:

We want to find the cost (x ) for 100 sachets:

Gh¢5/30 sachets = Gh¢x/100 sachets

Step 3: Using cross-multiplication solve for x :

5 × 100 = 30 × x 500 = 30x Divide both sides by 30 to isolate x :

x = 500/30 x = 16.67 Therefore, the cost of 100 sachets of water would be Gh¢16.67.

Example 5.12

A model of the Kantanka Onantefo SUV is built to a scale of 1:18.

If the actual length of the SUV is 4.5 metres, how long is the model SUV?

Solution

Step 1: Understand the scale ratio:

The scale of the model is 1:18, which means 1 unit of length on the model represents 18 units of the same length on the actual vehicle.

Step 2: Write down the given information:

Actual length of the SUV = 4.5 metres Scale = 1:18

Step 3: Set up a proportion:

The ratio of the model’s length to the actual length is equal to the scale:

Model lenth_ Actual length = 1/18 Let the model length be × (in metres). Substituting the actual length:

x_ 4.5 = 1/18

Step 4: Solve for x: Cross-multiply to eliminate the fraction:

x × 18 = 4.5 × 1 18x = 4.5 Divide both sides by 18 to isolate x x = 4.5/18 = 0.25 Therefore, the model SUV is 0.25 metres (= 25 cm) long

Now let’s explore proportions which involve comparing two ratios or rates in depth. For example, when calculating the speed of a vehicle, we typically compare the distance covered with the time it takes. Equivalent proportions are essentially the same, even though they may appear slightly different in form.

You can determine if proportions are equivalent by verifying if their ratios are equal. Examples of this can be observed in geometric shapes like similar triangles.

Ratios of Angles in Triangles

When discussing ratios in angles, we often refer to the relationships between the angles of a triangle. The ratios can help us understand how the angles compare with each other and can be useful in solving problems involving triangles.

Understanding the application of ratios to angles in triangles is crucial as it lays the foundation for various mathematical concepts and real-world applications.

Pythagoras’ Theorem

This fundamental concept in geometry states that, in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Figure 5.3: Representation of Pythagoras’s theorem Mathematical Representation of Pythagoras’s theorem from the figure above is:

a² + b² = c² Where:

a and b are the lengths of the two sides that are adjacent to the right angle c is the length of the hypotenuse (the side opposite the right angle and the longest side) For example: In a right-angled triangle with sides of length 3, 4 and 5:

3² + 4² = 5² 9 + 16 = 25 25 = 25 This demonstrates that Pythagoras’ Theorem holds true for this triangle and will hold true for all right-angled triangles.

Remember too that all that lovely trigonometry is all about trigonometric ratios.

Where, in a right-angled triangle the following always holds true with these ratio:

sinθ = Opposite________ Hypotenuse cosθ = Adjacent________ Hypotenuse tanθ = Opposite________ Adjacent

Example 5.13

If the interior angles of a triangle are 45°, 45° and 90°:

1. What is the ratio of its interior angles?

2. What is the ratio of its side lengths?

Solution

1. The ratio of the interior angles is: 45 : 45 : 90, which simplifies to 1 : 1 : 2.

2. The ratio of its side lengths is: 1 : 1 : √_ 2, as the triangle must be isosceles so we can make:

a = b = 1 and c = √1²+ 1²= √2 Everyday uses of proportions Proportions help us solve problems in everyday life when one part of a relationship is unknown.

Let us explore the practical applications of proportions in our daily lives through the following examples.

Example 5. 14

A kente cloth weaver wants to create a traditional kente design by combining red and yellow reels.

If they use 20 reels of red, how many threads of yellow should they add to maintain a 4:3 ratio of red to yellow?

Solution

Step 1: Write down the ratio: Red : Yellow = 4:3

Step 2: Identify the number of red threads: Number of red threads = 20

Step 3: Set up a proportion:

Since the ratio of red to yellow is 4:3, we can set up a proportion with the number of red threads:

20_ x = 4/3

Step 4: Solve for x :

To solve for x , we can cross-multiply:

20 × 3 = 4 × x 60 = 4x Now, divide both sides by 4:

x = 60/4 x = 15 Therefore, the kente cloth weaver should add 15 reels of yellow to maintain a 4:3 ratio of red to yellow.

Alternative Method

Table 5.2: Ratios

Parts (Kente) Red reels Yellow reels Ratio Total 1ˢᵗ4 3 4:3 7 2ⁿᵈ8 6 8:6 14 3ʳᵈ12 9 12:9 21 4ᵗʰ16 12 16:12 28 5ᵗʰ20 15 20:15 35 We notice from the table above that for every 20 of red reels there is a corresponding 15 of yellow thread. Thus, 4 : 3 = 20 : 15, so 15 yellow reels are needed.

Example 5.15

A caterer preparing for the Homowo festival wants to make traditional kpokpoi in a 4:1 ratio of cornmeal to palm nut soup.

If they have 16 cups of cornmeal, how many cups of palm nut soup do they need to keep the ratio?

Solution

Step 1: Write down the ratio: Cornmeal : Palm nut soup = 4:1

Step 2: Identify the number of cups of cornmeal: Number of cups of cornmeal = 16

Step 3: Set up a proportion:

Since the ratio of cornmeal to palm nut soup is 4:1, we can set up a proportion.

16_ x = 4/1

Step 4: Solve for x :

Cross-multiply:

16 × 1 = 4 × x 16 = 4x Now, divide both sides by 4:

x = 16/4 x = 4 Therefore, the caterer needs 4 cups of palm nut soup to keep the 4:1 ratio of cornmeal to palm nut soup.

Distance-time graphs, also known as travel graphs, are graphical representations that illustrate the relationship between the distance travelled by an object or person and the time taken to cover that distance.

A distance-time graph typically displays:

1. Distance (in units such as metres, kilometres, miles, etc.) on the vertical axis (y-axis)

2. Time (in units such as seconds, minutes, hours, etc.) on the horizontal axis (x-axis)

Figure 5.4: Features of a distance-time graph The implications of the gradient of a distance time graph

1. In a distance-time graph, the gradient (or slope) of the line indicates the speed or velocity of the object.

2. It illustrates the rate at which distance changes in relation to time.

3. Mathematically, gradient = rise____ run = change in distance_____________ change in time

4. A steeper gradient suggests that the object covers more distance in a shorter amount of time, indicating a higher speed as illustrated by the 30mph line in the figure above.

5. A shallower gradient means the object covers less distance over a longer period, reflecting a slower speed as illustrated by the 20mph line above.

6. A horizontal line (with zero gradient) shows no change in distance over time, meaning the object is at rest as illustrated by the flat lines above.

7. Note that from 15:00 to 16:00 hours the object is returning, at a constant speed of 40mph, to the starting point. This means that the distance represented on the y-axis is the distance from a fixed start point.

8. Straight lines represent constant speed or velocity. If the lines are curved, concave or convex, this represents acceleration or deceleration respectively, as shown below.

Figure 5.5: Acceleration and deceleration Working in pairs, do the following activity to understand how to plot a distance time graph.

Activity 5.5: Understanding how to plot a distance-time graph and calculate average speed.

Materials:

• Graph paper

• Pencil

• Ruler

• Calculator Instructions:

1. Draw your axes:

The x-axis represents time in hours and should go from 0 to 3 hours.

The y-axis represents distance in kilometres and should go from 0 to 150 kms.

2. Plot the points: On the graph paper, plot the points (0, 0) and (3, 150) to represent the distance travelled by a bus over time.

3. Draw the line: Draw a straight line through the two points to represent the distance-time graph. This shows a steady speed.

4. Calculate the average speed: Use the formula:

Average speed = Total distance___________ Total time In this case, the total distance is 150 km and the total time is 3 hours.

5. Record the average speed: Write down the average speed on a piece of paper.

6. Analyse the graph: Look at the distance-time graph and describe how it appears.

It should be a straight line with a positive gradient showing a steady speed

Example 5.16

A cyclist covers a distance of 36 km in 1.5 hours.

What is the average speed and how would this appear on a distance-time graph?

Solution

Step 1: Write down the given information: Distance covered = 36 km Time taken = 1.5 hours

Step 2: Recall the formula for average speed: Average Speed = Total Distance____________ Total Time

Step 3: Substitute the values into the formula: Average Speed = 36 km/1.5 hours Therefore, the average speed = 24 km/h On a distance-time graph, this would appear as a straight line with a gradient (or slope) representing the average speed of 24 km/h. The line would start at the origin (0 km, 0 hours) and end at the point (36 km, 1.5 hours).

Example 5.17

On a distance-time graph, a line goes from (0 hours, 0 km) to (4 hours, 200 km).

What is the speed of the object?

Solution

Calculate Speed:

Speed = Total distance___________ Total time Speed = 200 km/4 hours = 50 km/h.

Therefore, the speed of the object is 50 km/h.

Example 5.18

A swimmer covers a distance of 1 500 metres in 25 minutes.

What was their speed in m/s?

Solution

Step 1: Write down the given information:

Distance covered = 1 500 metres Time taken = 25 minutes

Step 2: Convert the time from minutes to seconds:

Since there are 60 seconds in a minute, multiply the time in minutes by 60:

25 minutes × 60 = 1 500 seconds

Step 3: Recall the formula for speed:

Speed = Distance_______ Time

Step 4: Substitute the values into the formula:

Speed = 1,500 metres/1,500 seconds Speed = 1 m/s Therefore, the swimmer’s speed is 1m/s

Ratios, rates and proportions are fundamental tools in financial mathematics, health, sharing of properties, sports, utility bills, Currency Exchange, etc.

Let us examine how these concepts are applied in real-life scenarios.

Everyday Applications

Example 5.19

A nutritionist recommends a ratio of 50% carbohydrates, 25% protein and 25% fat in a diet plan. If a person aims to consume 2 200 worth of calories per day, how many calories should come from carbohydrates?

Solution

To find the number of calories from carbohydrates:

Total daily calories = 2 200 Percentage of carbohydrates = 50% = 0.5 Calories from carbohydrates = Total daily calories × Percentage of carbohydrates = 22 00 × 0.5 = 1100 So, the person should allocate 1100 calories from carbohydrates.

Example 5.20

A different dietitian suggests a ratio of 60% carbohydrates, 20% protein and 20% fat for a healthy eating plan. If someone is budgeting 2 500 for their daily caloric intake, how many calories should come from protein?

Solution

Step 1: Write down the given information Total daily calories = 2 500 Percentage of protein = 20% = 0.2

Step 2: Recall the formula to find the calories from protein:

Calories from protein = Total daily calories × Percentage of protein

Step 3: Substitute the values into the formula:

Calories from protein = 2 500 × 0.2

Step 4: Calculate the calories from protein:

Calories from protein = 500 Therefore, the amount of calories from protein should be 500.

Example 5.21

A health coach advises a ratio of 45% carbohydrates, 35% protein and 20% fat in a meal plan.

If an individual is planning to use1800 calories each day, how many calories should come from fat?

Solution

Step 1: Write down the given information:

Total daily calories = 1800 Percentage of fat = 20% = 0.2

Step 2: Recall the formula to find the calories from fat:

Calories from fat = Total daily calories × Percentage of fat

Step 3: Substitute the values into the formula:

Calories from fat = 1800 × 0.2

Step 4: Calculate the calories from fat:

Calories from fat = 360 Therefore, the amount of calories from fat should be 360.

Blood Pressure and Heart Rate Ratios

Example 5.22

A healthy resting blood pressure is typically within the range of 90 − 120 mmHg (systolic) and 60 − 80 mmHg (diastolic).

If someone’s resting systolic blood pressure is 110 mmHg, what is a desirable range for their heart rate?

Generally, a good rule of thumb is that diastolic pressure should be around two- thirds of heart rate plus + 2

Solution

Diastolic Pressure = 2/3 × Heart Rate + 2

Step 1: Determine the desirable diastolic pressure range:

The typical diastolic pressure range is 60 – 80 mmHg

Step 2: Rearrange the formula to solve for heart rate:

Heart rate = Diastolic Pressure − 2________________ 2/3 Heart Rate=1.5 × (Diastolic Pressure − 2)

Step 3: Calculate the heart rate for each extreme value in the diastolic range:

For a diastolic pressure of 60 mmHg:

Heart Rate = 1.5 × (60 − 2) = 1.5 × 58 = 87bpm For a diastolic pressure of 80 mmHg:

Heart Rate = 1.5 × (80 − 2) = 1.5 × 78 = 117 bpm Therefore, the desirable range for heart rate is between about 87 bpm to 117 bpm Calculating Body Mass Index (BMI) Body Mass Index (BMI) is a widely used measurement to assess an individual’s weight status in relation to their height. The formula is BMI = weight(kg)________ height(m )².

After calculating the BMI, we compare it to the BMI Categories − Und erweight: < 18.5 − Norma l weight: 18.5 − 24.9 − Overweight: 25 − 29.9 − Ob es e: ≥ 30

Note, these must be taken with caution as there are many more factors at play here.

Example 5.23

Niyil weighs 85 kg and has a height of 1.8 metres.

Calculate his Body Mass Index (BMI)

Solution

BMI = weight(kg)________ height(m )²Weight = 85 kg Height = 1.8 m BMI = 85_____ (1.8 )²= 85/3.24 ≈ 26.23 Niyil’s BMI is approximately 26.23

Example 5.24

Yamba Bernice weighs 60 kg and has a height of 1.5 metres.

Calculate her Body Mass Index (BMI)

Solution

BMI = weight(kg)________ height(m )²Weight = 60 kg Height = 1.5 m BMI = 60_____ (1.5 )²= 60/2.25 ≈ 26.67 Yamba Bernice’s BMI is approximately 26.67 Undertake the activity below on Calculating BMI and knowing the category it falls

Activity 5.6: Calculating Your BMI and Category

Step 1: Measure Your Weight and Height:

- Measure your weight in kilograms (kg) using a weight scale.

- Measure your height in metres (m) using a height measuring tape or stadiometre.

Step 2: Record Your Measurements:

- Write down your weight in kg: _____

- Write down your height in m: _____

Step 3: Calculate Your BMI:

- Use the formula: BMI = weight (in kg)_____________ height (in meters)²

- Substitute your values: BMI = _____ kg / (_____ m)²

- Calculate your BMI: _____

Step 4: Determine Your BMI Category:

- Use the BMI category chart below:

- Underweight: BMI < 18.5

- Normal weight: BMI = 18.5-24.9

- Overweight: BMI = 25-29.9

- Obese: BMI ≥ 30

- Determine which category you fall into: _____

Step 5: Record Your Results:

- Write down your BMI and category: BMI = ____, Category:

However, please note that there are a lot of caveats around this so please do take your category with caution.

Example 5.25

A patient weighing 50 kg is prescribed amoxicillin at a dose of 10 mg/kg/day, to be given in 2 divided doses. How many milligrams of amoxicillin should be given per dose?

Solution

Determine the total daily dose: The prescription states 10 mg/kg/day.

For a patient weighing 50 kg: Total daily dose = 10mg / kg / day × 50kg = 500mg / day.

Divide the total daily dose by the number of doses per day: The medication is to be given in 2 divided doses:

Dose per administration = Total daily dose_________________ Numner ofdoses per day = 500/2 = 250mg The patient should receive 250 mg of amoxicillin per dose.

Example 5.26

A man is 175 cm tall and weighs 82 kg. Calculate his BMI and determine which BMI category he falls into.

Solution

The formula is BMI = weight(kg)________ height(m )²Convert height to metres 175 cm = 1.75 m BMI = 82______ (1.75 )²= 82/3.0625 ≈ 26.8 Therefore. the man’s BMI is 26.8 kg/m².

Comparing with BMI Categories

− Und erweight: < 18.5 − Norma l weight: 18.5 − 24.9 − Overweight: 25 − 29.9 − Ob es e: ≥ 30 The man’s BMI places him in the Overweight category.

Division of Assets Among Heirs

Example 5.27

A will states that a property valued at Gh¢ 250 000 should be divided equally among 5 siblings. How much will each sibling receive?

Solution

Total value of property = Gh¢ 250 000 Number of siblings = 5 Amount per sibling = Total value of property ÷ Number of siblings = Gh¢ 250 000 ÷ 5 = Gh¢ 50 000 Each sibling will receive Gh¢ 50 000.

Partnership Profit Sharing

Example 5.28

Two partners agree to share profits in a 4:1 ratio.

If the total profit for the year is GH¢75 000, how much will each partner receive?

Solution

Calculate the total parts in the ratio: 4 + 1 = 5 parts Then, divide the total profit by the total parts:

Total profit = Gh¢75,000 Total parts = 5 Profit per part = Total profit ÷ Total parts = Gh¢75 000 ÷ 5 = Gh¢15 000 Now, multiply the profit per part by each partner’s parts:

Partner 1 (4 parts): 4 × Gh¢15 000 = Gh¢60 000 Partner 2 (1 part): 1 × Gh¢15 000 = Gh¢15 000 So, Partner 1 will receive Gh¢60 000 and Partner 2 will receive Gh¢15 000.

Example 5.29

Three partners agree to share profits in a 5:3:2 ratio.

If the total profit for the year is Gh¢90 000, how much will each partner receive?

Solution

5 + 3 + 2 = 10 parts in total Divide the total profit by the total parts:

Total profit = Gh¢90 000 Total parts = 10 Profit per part = Total profit ÷ Total parts = Gh¢90 000 ÷ 10 = Gh¢9 000 Now, multiply the profit per part by each partner’s parts:

Partner 1 (5 parts): 5 × Gh¢9 000 = Gh¢45 000 Partner 2 (3 parts): 3 × Gh¢9 000 = Gh¢27 000 Partner 3 (2 parts): 2 × Gh¢9 000 = Gh¢18 000 So, Partner 1 will receive Gh¢45 000, Partner 2 will receive Gh¢27 000 and Partner 3 will receive Gh¢18 000 Land Area Distribution

Example 5.30

A farmer has 200 hectares of land and wants to distribute it among his 3 daughters in a 2:3:5 ratio. How much land will each daughter receive?

Solution

2 + 3 + 5 = 10 parts in total Divide the total land by the total parts:

Total land = 200 hectares Total parts = 10 Land per part = Total land ÷ Total parts = 200 hectares ÷ 10 = 20 hectares Now, multiply the land per part by each daughter’s parts:

Daughter 1 (2 parts): 2 × 20 hectares = 40 hectares Daughter 2 (3 parts): 3 × 20 hectares = 60 hectares Daughter 3 (5 parts): 5 × 20 hectares = 100 hectares So, Daughter 1 will receive 40 hectares, Daughter 2 will receive 60 hectares and Daughter 3 will receive 100 hectares.

Investment Portfolio Allocation

Example 5.31

An investor has a portfolio of Gh¢200 000 and wants to allocate it with 70% in stocks and 30% in bonds.

How much should be invested in each asset class?

Solution

Given:

• Total Portfolio = Gh¢200 000

• Stocks = 70%

• Bonds = 30%

Step 1: Calculate the amount for stocks:

Amount in stocks = 70 % × 200 000 = 70/100 × 200 000 = Gh¢140 000

Step 2: Calculate the amount for bonds:

Amount in bonds = 30 % × 200 000 = 30/100 × 200 000 = Gh¢60 000 Therefore, the amount that should be invested in Stocks is Gh¢140 000 and Bonds is Gh¢60 000 Shooting Percentages in Sports

Example 5.32

A soccer player takes 20 penalty kicks during a season and successfully scores 15 of them.

What is their scoring percentage for the season?

Solution

Successful penalty kicks = 15 Total penalty kicks attempted = 20 Scoring percentage = Successful penalty kicks_____________________ Total penalty kicks attempted × 100 = 15/20 × 100 = 0.75 × 100 = 75% The player’s scoring percentage for the season is 75% Currency Exchange Currency Exchange refers to the process of converting one country’s currency into another country’s currency. This is typically done based on the current exchange rate, which is the value of one currency in relation to another.

Example 5.33

The exchange rate between the Ghanaian cedi (Gh¢) and the US dollar (USD) is 1 USD = Gh¢. 12.50 How much would $100 USD is in cedis?

Solution

To find the equivalent amount in Ghanaian cedis, multiply the amount in US dollars by the exchange rate:

$100 USD × 12.50 Gh¢/USD = Gh¢1 250 So, $100 USD is equivalent to Gh¢1 250 Commission Rates for Transactions

Example 5.34

When exchanging currency, there might be a commission fee. If you’re exchanging €200 (EUR) and the commission rate is 3%, what is the total cost in EUR?

Solution

To find the total cost, calculate the commission fee and add it to the original amount:

Commission rate = 3% = 3/100 = 0.03 Original amount = €200 Commission fee = Original amount × Commission rate = €200 × 0.03 = €6 Total cost = Original amount + Commission fee = €200 + €6 = €206 Therefore, the total cost is €206.

Purchasing Power Comparisons

Purchasing power refers to the amount of goods and services that can be bought with a currency in a particular place. Even if exchange rates are similar, the cost of living can differ from one country to another.

For example: In Ghana, Gh¢10 might be enough to buy a loaf of bread, while in the US, the same loaf could cost $2 USD, which is equivalent to Gh¢20.40 at the current the exchange rate. Although the exchange rate makes the cedi appear stronger, the lower cost of living in Ghana means that $2 USD would go further in Ghana than in the US.

Currency Pair Ratios in Forex Trading

Forex traders predict how currency exchange rates will change in the future. They often use currency pair ratios to compare the value of two currencies.

For example, the EUR/USD pair represents the Euro against the US Dollar. If the ratio is 1.20, it means 1 Euro costs 1.20 US Dollars. Traders analyse different factors to anticipate whether this ratio will rise (indicating a stronger Euro against the Dollar) or fall (indicating a stronger Dollar against the Euro), which influences their potential gains or losses.

Utility Bills

Activity 5.7: Electricity consumption project Objective:

Calculate and report the average daily electricity consumption of your family for a month.

Follow the instructions to carry out electricity consumption project:

Step 1: Gather Information: Collect your family’s electricity bills for the past month.

Step 2: Record Data: Note down the total kilowatt-hours (kWh) consumed for the month.

Step 3: Calculate Daily Average: Divide the total kWh consumed by the number of days in the month to find the average daily consumption.

Step 4: Additional Observations: Take note of any factors that might have affected your family’s electricity consumption, such as changes in weather, number of people at home, or new appliances.

Step 5: Report Findings: Prepare a short report or presentation to share with the class, including:

1. Total monthly electricity consumption (kWh)

2. Average daily electricity consumption (kWh)

3. Any notable observations or factors affecting consumption This project aims to help you understand your family’s electricity consumption patterns and encourage responsible energy usage.

Drug Dosage Calculations

Drug dosage calculations are crucial in healthcare to ensure patients receive the correct amount of medication.

Common dosage calculation formulas Dos a ge = (Pres crib ed d os e × Pa tient weight)__________________________ S ta nd a rd d os e Drip ra te (mL/hr) = (Vol ume × Drip fa ctor)__________________ Time Dose per kilogram: Dose (mg/kg) = (Desired dose × Weight in kg)______________________ Dose on hand Dose per pound: Dose (mg/lb) = (Desired dose × Weight in lb)_____________________ Dose on hand Volume to be administered: Volume (mL) = (Dose × Volume of solution)_____________________ Strength of solution In real-world healthcare settings, accurate dosage calculations play a critical role in patient care.

Specifically:

1. Nurses rely on these calculations daily to administer medications safely and effectively.

2. Pharmacists verify dosage calculations to ensure accuracy and prevent medication errors.

3. Doctors prescribe medications based on these calculations, taking into account individual patient factors and medication ratios.

1. In a pet shop in Accra, the ratio of kittens to puppies is 4:6. If there are 60 animals in total, how many kittens are there?

2. A trotro travels from Accra to Kumasi, covering a distance of 300 kilometres in 5 hours. What is its average speed in kilometres per hour?

3. If 2 kg of oranges cost Gh¢10.00, what is the price per kilogram of oranges?

4. In a class of 25 students in Shalom Star Academy in Chinderi, the ratio of boys to girls is 3:2. How many girls are in the class?

5. If 8 masons can complete a building project in 15 days, how many days would it take 20 masons to complete the same project?

6. A recipe for kelewele requires 3 cups of flour for every 4 ripe plantains. If you want to use 8 ripe plantains, how many cups of flour do you need?

7. If a 15-metre shadow is cast by a 25-metre-tall cocoa tree, how tall is a mango tree that casts a 10-metre shadow at the same time of day?

8. A university randomly selected students to take a survey about their preferred mode of transportation. Of the students selected for the survey, 18 were first-year students and 24 were continuing students. Find the ratio that represents:

a. part-to-part: first years : continuing students

b. part-to -whole: first years : all students selected

9. A map has a scale of 1:50 000. If the distance between Accra and Tema is represented as 8 cm on the map, what is the actual distance in kilometres?

10. If 5 pairs of sandals cost Gh¢75, how much would 9 pairs of sandals cost at the same rate?

11. A solution is 15% salt. How many millilitres of salt are in 200 ml of the

solution?

12. A school has 400 students, and the student-to-teacher ratio is 20:1. How many teachers are there?

13. What type of line would represent an object traveling at a steady velocity on a distance-time graph?

14. What does a concave-upward curve on a distance-time graph typically signify?

15. What event is represented when two or more lines meet at a single point on a distance-time graph?

16. What shape or feature on a distance-time graph indicates that an object is slowing down?

17. What physical quantity is represented by the area beneath the curve on a distance-time graph?

18. Describe the shape of a distance-time graph for an object that travels away from a reference point, reverses direction and returns to the starting point.

19. How would you represent a trotro that travels at a steady speed from Accra to Tema, stops for a break and then continues at a faster speed to Kumasi on a distance-time graph?

20. A bus accelerates from rest to 25 m/s over 15 seconds. What is its acceleration, and how would this be represented on a distance-time graph?

21. A distance-time graph shows a line from (0s, 0m) to (4s, 40m), then a horizontal line to (8s, 40m). What was the object’s average speed over the entire 8 seconds?

22. A motorbike travels 25 km east in 20 minutes, then returns 15 km west in 15 minutes. Calculate the total distance traveled and the displacement.

How would this journey appear on a distance-time graph?

23. If the ratio of the sides of a right-angled triangle is 15: 20: x, where × is the hypotenuse. Find the missing length x and use these values to establish the interior angles of the triangle, to the nearest degree.

24. A basketball player takes 8 free throws during a game. They successfully make 5 of them. What is their success rate (win-loss ratio) expressed as a decimal for free throws during this game?

25. Player × has an efficiency rating of 102, while Player Y has a rating of 95.

Based on this rating system, which player is considered statistically more successful?

26. The exchange rate between the US dollar (USD) and the Japanese Yen (JPY) is 1 USD = 150 JPY. You have $300 USD and want to convert it to Yen.

How many Yen will you receive?

27. You’re exchanging 100 British Pounds (GBP) for Ghanaian Cedis (Gh¢).

The exchange rate is 1 GBP = Gh¢12.00, but there›s a 2% commission fee. What is the total cost of the transaction in British Pounds?

28. Imagine a sandwich costs 4 British Pounds in the UK and 30 Brazilian Reais (BRL). The exchange rate is 1 GBP = 5 BRL. In which country would you get more sandwiches for your money (considering exchange rates)? Explain your answer.

29. Your water bill shows you consumed an average of 20 cubic metres (m³) of water per day for the past month. How many cubic metres did you consume in total for the entire month (assuming 30 days)?

30. 30.Your electricity bill uses a tiered pricing structure. The first 300 kWh of electricity costs 0.12 Gh¢ per kWh, but the rate increases to Gh¢0.18 per kWh for any consumption exceeding 300 kWh. If you used a total of 350 kWh last month, what was your total electricity bill in Gh¢?