WORD PROBLEMS ON ALGEBRAIC FRACTION

Subject : Mathematics

Class : JSS 2

Term : Second Term

Week : Week 2

Topic :

WORD PROBLEMS ON ALGEBRAIC FRACTION

Previous Lesson :

REVIEW OF FIRST TERM WORK, EXPANDING AND FACTORIZING ALGEBRAIC EXPRESSIONS, SOLVING OF QUADRATIC EQUATIONS.

Objective: Students will be able to solve word problems involving algebraic fractions.

Materials:

• Whiteboard and markers
• Printed handouts of algebraic fraction word problems
• Calculators (optional)
• Online Resources 

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Content ;

1. A contractor can finish a certain construction project alone in 10 days. Another contractor can finish the same project alone in 15 days. If they work together, how many days will it take them to complete the project?

Solution:

Let x be the number of days it takes for both contractors to finish the project.

The fraction of the project that the first contractor can finish in one day is 1/10. The fraction of the project that the second contractor can finish in one day is 1/15. If they work together, the fraction of the project they can finish in one day is:

1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6

Therefore, the equation we need to solve is:

1/10 + 1/15 = 1/x

Multiplying both sides by 30x, we get:

3x + 2x = 30

5x = 30

x = 6

Therefore, it will take both contractors 6 days to complete the project working together.

2. A recipe for a cake calls for 1/4 cup of sugar. If you want to make half the recipe, how much sugar should you use?

Solution:

If we want to make half the recipe, we need to multiply all the ingredients by 1/2. Therefore, the amount of sugar we need is:

1/4 x 1/2 = 1/8

Therefore, we need 1/8 cup of sugar to make half the recipe.

3. A certain investment yields 6% interest per year. If you invest ₦5000 for one year and ₦8000 for another year, what is the average annual interest rate for the two investments?

Solution:

The interest earned from the first investment is:

₦5000 x 6% = ₦300

The interest earned from the second investment is:

₦8000 x 6% = ₦480

The total interest earned is:

₦300 + ₦480 = ₦780

The total amount invested is:

₦5000 + ₦8000 = ₦13000

The average annual interest rate is:

₦780/₦13000 x 100% = 6%

Therefore, the average annual interest rate for the two investments is 6%.

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More Examples

1. John and Mary can paint a room together in 6 hours. If John can paint the same room alone in 8 hours, how long would it take Mary to paint the room alone?

Solution:

Let x be the time it takes for Mary to paint the room alone.

The fraction of the room John can paint in one hour is 1/8. If they work together, the fraction of the room they can paint in one hour is:

1/6 = (1/8) + (1/x)

Multiplying both sides by 24x, we get:

4x = 3x + 24

x = 24

Therefore, it would take Mary 24 hours to paint the room alone.

2. A train travels from Station A to Station B, a distance of 400 km, in 4 hours. On the return trip, the train travels from Station B to Station A at a speed 25% slower than its speed on the first trip. If the return trip takes 5 hours, what is the speed of the train on the first trip?

Solution:

Let s be the speed of the train on the first trip.

The time it takes for the train to travel from Station A to Station B is:

4 = 400/s

s = 100 km/h

On the return trip, the speed of the train is 25% slower, or 75 km/h.

The time it takes for the train to travel from Station B to Station A is:

5 = 400/75

Therefore, the speed of the train on the first trip is 100 km/h.

3. A recipe for a fruit salad calls for 2/3 cup of strawberries, 1/2 cup of blueberries, and 1/4 cup of chopped pineapple. If you want to make twice the recipe, how much of each ingredient should you use?

Solution:

To make twice the recipe, we need to multiply each ingredient by 2.

The amount of strawberries we need is:

2/3 x 2 = 4/3

Therefore, we need 4/3 cups of strawberries.

The amount of blueberries we need is:

1/2 x 2 = 1

Therefore, we need 1 cup of blueberries.

The amount of pineapple we need is:

1/4 x 2 = 1/2

Therefore, we need 1/2 cup of chopped pineapple

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Algebraic Processes in Mathematics

Evaluation

1. Simplify the expression: (3x^2 + 2x – 1)/(x^2 – 1) a) 3x + 2 b) 3x – 2 c) 3x + 1 d) 3x – 1
2. Simplify the expression: (x^2 + 3x + 2)/(x^2 – 4x + 3) a) (x + 2)/(x – 1) b) (x + 1)/(x – 3) c) (x + 3)/(x – 1) d) (x + 2)/(x – 3)
3. Solve for x: (x/3) + 1 = (5x + 2)/(6x) a) x = -1/7 b) x = 1/7 c) x = 2/7 d) x = -2/7
4. Simplify the expression: (2x^2 – 6x + 4)/(x^2 – 4) a) (2x – 4)/(x + 2) b) (2x + 4)/(x – 2) c) (2x – 4)/(x – 2) d) (2x + 4)/(x + 2)
5. Solve for x: (x + 1)/(x – 2) – (2x – 1)/(x + 1) = 0 a) x = -1/2 b) x = -1 c) x = 1/2 d) x = 1
6. Simplify the expression: (2x^2 – 3x – 5)/(x^2 – 4x + 4) a) (2x – 5)/(x – 2) b) (2x + 5)/(x – 2) c) (2x – 5)/(x – 4) d) (2x + 5)/(x – 4)
7. Solve for x: (x^2 – 6x + 8)/(x^2 – x – 12) = 2/3 a) x = -2, 4 b) x = 2, 4 c) x = -2, -4 d) x = 2, -4
8. Simplify the expression: (x^2 – 9)/(x^2 + 2x – 15) a) (x – 3)/(x – 5) b) (x + 3)/(x + 5) c) (x – 3)/(x + 5) d) (x + 3)/(x – 5)
9. Solve for x: (3x – 1)/(x + 1) – (x – 2)/(x – 1) = 2 a) x = -1/2 b) x = 1/2 c) x = 2/3 d) x = -2/3
10. Simplify the expression: (4x^2 – 16)/(x^2 – 5x – 6) a) (4x + 4)/(x – 6) b) (4x – 4)/(x + 6) c) (4x + 4)/(x + 6) d) (4x – 4)/(x – 6)

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Class work

1. If 2/5 of a number is equal to 1/3 of another number, what is the ratio of the first number to the second number? a) 3:10 b) 5:6 c) 6:5 d) 10:3
2. If the sum of two numbers is 9 and their ratio is 2:3, what is the larger number? a) 3 b) 4 c) 5 d) 6
3. A pipe can fill a tank in 6 hours, while another pipe can fill the same tank in 4 hours. How long will it take both pipes to fill the tank together? a) 1.5 hours b) 2 hours c) 3 hours d) 4 hours
4. If 1/3 of the workers in a factory are women and 3/5 of the women are married, what fraction of the workers in the factory are married women? a) 1/5 b) 1/3 c) 1/4 d) 1/2
5. A recipe for a cake calls for 2/3 cup of sugar. If you want to make half the recipe, how much sugar should you use? a) 1/4 cup b) 1/3 cup c) 1/2 cup d) 1 cup
6. John and Mary can paint a room together in 6 hours. If John can paint the same room alone in 8 hours, how long would it take Mary to paint the room alone? a) 12 hours b) 16 hours c) 24 hours d) 32 hours
7. If the sum of two numbers is 10 and their difference is 2, what is their product? a) 18 b) 20 c) 24 d) 28
8. A car travels 120 km in 2 hours and then travels 180 km in 3 hours. What is the average speed of the car? a) 60 km/h b) 65 km/h c) 70 km/h d) 75 km/h
9. A shopkeeper sells 1/4 of his stock on the first day, 1/3 of the remaining stock on the second day, and the rest on the third day. What fraction of his stock is sold on the third day? a) 1/6 b) 1/4 c) 1/3 d) 1/2
10. A boat can travel 15 km/h in still water. If it can travel 4 km upstream in the same amount of time as it can travel 10 km downstream, what is the speed of the current? a) 3 km/h b) 4 km/h c) 5 km/h d) 6 km/h

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EQUATIONS WITH FRACTIONS

Lesson Presentation

Revision

• Revise the last topic which was REVIEW OF FIRST TERM WORK, EXPANDING AND FACTORIZING ALGEBRAIC EXPRESSIONS, SOLVING OF QUADRATIC EQUATIONS.

Introduction (5 minutes):

• Begin by asking students if they have ever encountered word problems involving algebraic fractions before.
• Briefly review the concept of algebraic fractions and how they differ from regular fractions.
• Explain that in this lesson, they will learn how to apply algebraic fractions to solve real-life word problems.

Activity 1: Examples and Practice (20 minutes):

• Provide the students with several examples of word problems involving algebraic fractions on the whiteboard.
• Work through the examples together as a class, showing step-by-step how to identify the problem, set up the equation, and solve for the variable.
• After working through several examples, give the students handouts with additional practice problems to work on independently or in pairs.
• Circulate around the classroom to answer any questions and provide assistance as needed.

Activity 2: Group Work (15 minutes):

• Divide the students into small groups of 3-4.
• Provide each group with a set of word problems to solve.
• Instruct the groups to work together to solve the problems, showing their work on paper and checking their answers with calculators or each other.
• After a designated amount of time, bring the groups back together and review the answers together as a class.

Evaluation 

1. If 2/5 of a number is equal to 1/3 of another number, the ratio of the first number to the second number is ___________.
2. If the sum of two numbers is 9 and their ratio is 2:3, the larger number is ___________.
3. A pipe can fill a tank in 6 hours, while another pipe can fill the same tank in 4 hours. Together, both pipes can fill the tank in ___________ hours.
4. If 1/3 of the workers in a factory are women and 3/5 of the women are married, the fraction of the workers in the factory that are married women is ___________.
5. A recipe for a cake calls for 2/3 cup of sugar. If you want to make half the recipe, you need ___________ cup(s) of sugar.
6. John and Mary can paint a room together in 6 hours. If John can paint the same room alone in 8 hours, Mary can paint the room alone in ___________ hours.
7. If the sum of two numbers is 10 and their difference is 2, their product is ___________.
8. A car travels 120 km in 2 hours and then travels 180 km in 3 hours. The average speed of the car is ___________ km/h.
9. A shopkeeper sells 1/4 of his stock on the first day, 1/3 of the remaining stock on the second day, and the rest on the third day. The fraction of his stock sold on the third day is ___________.
10. A boat can travel 15 km/h in still water. If it can travel 4 km upstream in the same amount of time as it can travel 10 km downstream, the speed of the current is ___________ km/h

Conclusion (5 minutes):

• Recap the lesson by reviewing key concepts and strategies for solving word problems involving algebraic fractions.
• Encourage the students to continue practicing and applying these skills to future math problems.
• End the lesson by answering any final questions and thanking the students for their participation [the_ad id=”40091″]