Solving Equation Expressions – JSS3 Mathematics Lesson Note (Third Term)

Solving equation expressions means writing a word problem as an equation and finding the unknown number.

Behavioural objectives

By the end of the lesson, students should be able to:

1. form and solve equations from word problems;
2. clear fractions in an equation using the LCM of the denominators and solve it;
3. solve word problems on sums, differences and consecutive numbers;
4. solve word problems on products;
5. solve direct and inverse proportion problems by the unitary method;
6. calculate simple interest and compound interest.

How do you solve word problems?

To solve a word problem, let a letter stand for the unknown number, form an equation, then solve it.

• Equation: a statement that two expressions are equal, such as 5x = 20 – 2x.
• Unknown: the number we are asked to find, shown by a letter such as x or n.
• LCM: the smallest number that every denominator divides into exactly.

Steps for solving a word problem

1. Read the problem and decide what number is unknown.
2. Let a letter stand for the unknown number.
3. Change each phrase into symbols and form the equation.
4. Clear any fractions by multiplying each term by the LCM of the denominators.
5. Collect the letter terms on one side and the numbers on the other.
6. Solve for the letter, then check by putting the answer back into the problem.

Which words mean which operation?

Worked examples

1. Find 1/4 of the positive difference between 29 and 11.
2. The product of a certain number and 5 is equal to twice the number subtracted from 20. Find the number.
3. The sum of 35 and a certain number is divided by 4, and the result is equal to double the number. Find the number.

Solutions

1. Positive difference = 29 – 11 = 18
   1/4 of 18 = 18/4 = 4 1/2
2. Let the number be x.
   x × 5 = 20 – 2x
   5x = 20 – 2x
   5x + 2x = 20
   7x = 20
   x = 20/7 = 2 6/7
3. Let the number be n.
   Sum of 35 and n = n + 35
   Divided by 4 = (n + 35)/4
   Result = 2 × n = 2n
   Therefore (n + 35)/4 = 2n
   n + 35 = 8n
   8n – n = 35
   7n = 35
   n = 35/7 = 5

Practice questions

1. From 50 subtract the sum of 3 and 5, then divide the result by 6.
2. The sum of 8 and a certain number is equal to the product of the number and 3. Find the number.

What are the types of word problems?

Word equations or expressions can be grouped into three types: sums and differences, products, and expressions with fractions.

• Sums and differences: problems that add or subtract numbers, including consecutive numbers.
• Products: problems that multiply numbers together.
• Expressions with fractions: equations whose terms have denominators, solved by clearing the fractions first.

What are sums and differences?

The sum of a set of numbers is the result of adding them, and the difference between two numbers is the result of subtracting one from the other.

• Sum: the sum of -2 and -3 is -2 + (-3) = -5.
• Difference: the difference between 19 and 8 is 19 – 8 = 11.
• Positive difference: subtract the smaller number from the larger, so the answer is positive.

What are consecutive numbers?

Consecutive numbers are numbers that follow one after another in order, with the same gap between them.

• Consecutive numbers: 1, 2, 3, 4, 5, 6, … written as x, x + 1, x + 2, …
• Consecutive even numbers: 2, 4, 6, 8, 10, … written as 2x, 2x + 2, 2x + 4, … (or x, x + 2, x + 4, … where x is even).
• Consecutive odd numbers: 1, 3, 5, 7, 9, … written as 2x + 1, 2x + 3, 2x + 5, … (or x, x + 2, x + 4, … where x is odd).

Worked examples

1. Find the sum of -2 and -3.4.
2. Find the positive difference between 19 and 8.
3. The difference between two numbers is 7. If the smaller number is 7, find the other.
4. The difference between -3 and a number is 8. Find the two possible values for the number.
5. Find the three consecutive numbers whose sum is 63.

Solutions

1. -2 + (-3.4) = -5.4
2. 19 – 8 = +11
3. Let the other number be Y, that is Y – 7 = 7.
   Y = 7 + 7 = 14
4. Let m represent the number.
   m – (-3) = 8
   m + 3 = 8
   m = 8 – 3
   m = +5
   Also -3 – m = 8
   -m = 8 + 3
   -m = 11
   m = -11
   The possible values are +5 and -11.
5. Let the first number be x, the second x + 1 and the third x + 2.
   x + (x + 1) + (x + 2) = 63
   3x + 3 = 63
   3x = 63 – 3
   3x = 60
   x = 60/3 = 20
   The numbers are 20, 21 and 22.

Practice questions

1. Find the sum of all odd numbers between 10 and 20.
2. The sum of four consecutive odd numbers is 80. Find the numbers.
3. The difference between two numbers is 9 and the larger number is 32. Find the numbers.

What are products?

The product of two or more numbers is the result obtained when the numbers are multiplied together.

• Same signs: the product of two positive or two negative numbers is positive.
• Different signs: the product of a positive and a negative number is negative.
• Zero: any number multiplied by 0 gives 0.

Worked examples

1. Find the product of -6, 0.7 and 20/3.
2. The product of two numbers is 8. If one of the numbers is 1/4, find the other.
3. Find the product of the sum of -2 and 9 and the difference between -8 and -5.

Solutions

1. Product = -6 × 0.7 × 20/3
   = -6 × 7/10 × 20/3
   = (-6 × 7 × 20)/(10 × 3)
   = -2 × 7 × 2 = -28
2. Let the number be x.
   1/4 × x = 8
   Multiply both sides by 4.
   x = 8 × 4 = 32
3. Sum = -2 + 9 = 7
   Difference = -5 – (-8) = -5 + 8 = 3
   Product = 7 × 3 = 21

Practice questions

1. The product of three numbers is 0.084. If two of the numbers are 0.7 and 0.2, find the third number.
2. Find the product of the difference between 2 and 7 and the sum of 2 and 7.
3. Find the product of -4, 2.5 and 3.

How do you solve equations with fractions?

To solve an equation with fractions, first clear the fractions by multiplying each term by the LCM of the denominators.

Steps for clearing fractions

1. List the denominators of all the fractions.
2. Find their LCM.
3. Multiply every term on both sides by the LCM.
4. Cancel, then expand any brackets.
5. Collect like terms and solve for the letter.

Examples

Solve the following.

1. x/9 = 2
2. (x + 9)/5 + (2 + x)/2 = 0
3. 2x = (5x + 1)/7 + (3x – 5)/2

Solutions

1. x/9 = 2
   Multiply both sides by 9.
   x = 18
2. (x + 9)/5 + (2 + x)/2 = 0
   Multiply by the LCM (10).
   10 × (x + 9)/5 + 10 × (2 + x)/2 = 0 × 10
   2(x + 9) + 5(2 + x) = 0
   2x + 18 + 10 + 5x = 0
   2x + 5x + 28 = 0
   7x = -28
   x = -28/7 = -4
3. 2x = (5x + 1)/7 + (3x – 5)/2
   Multiply by the LCM (14).
   14 × 2x = 14 × (5x + 1)/7 + 14 × (3x – 5)/2
   28x = 2(5x + 1) + 7(3x – 5)
   28x = 10x + 2 + 21x – 35
   28x = 31x – 33
   28x – 31x = -33
   -3x = -33
   x = 33/3 = 11

Practice questions

Solve the following equations.

1. 7/3c = 21/2
2. 6/(y + 3) = 11/(y – 2)
3. 3/(2b – 5) – 4/(b – 3) = 0

What is proportion?

Proportion is the way two quantities change together, and it can be direct or inverse.

• Direct proportion: when one quantity increases, the other increases in the same ratio.
• Inverse proportion: when one quantity increases, the other decreases in the same ratio.
• Unitary method: find the value for one unit first, then find the value asked for.

Rules for the unitary method

1. Write in a sentence the quantity to be found at the end.
2. Decide whether the problem is direct or inverse proportion.
3. Find the rate for one unit before answering the problem.

How do direct and inverse proportion differ?

Direct proportion

Example: A worker gets ₦900 for 10 days of work. Find the amount for (a) 3 days, (b) 24 days, (c) x days.

Solution:

For 10 days = ₦900
For 1 day = 900/10 = ₦90

1. For 3 days = 3 × 90 = ₦270
2. For 24 days = 24 × 90 = ₦2,160
3. For x days = x × 90 = ₦90x

• Less time: 3 days gives less money, ₦270.
• More time: 24 days gives more money, ₦2,160.
• Conclusion: this is direct proportion.

Inverse proportion

Example 1: Seven workers dig a piece of ground in 10 days. How long will five workers take?

Solution:

For 7 workers = 10 days
For 1 worker = 7 × 10 = 70 days
For 5 workers = 70/5 = 14 days

Example 2: Five people took 8 days to plant 1,200 trees. How long will it take 10 people to plant the same number of trees?

Solution:

For 5 people = 8 days
For 1 person = 8 × 5 = 40 days
For 10 people = 40/10 = 4 days

Classwork

1. A woman is paid ₦750 for 5 days. Find her pay for (a) 1 day, (b) 22 days.
2. A piece of land has enough grass to feed 15 cows for x days. How long will it last (a) 1 cow, (b) y cows?
3. A bag of rice feeds 15 students for 7 days. How long would the same bag feed 10 students?

What is interest?

Interest is a payment given for saving or borrowing money, and it can be simple or compound.

• Principal (P): the money saved or borrowed at the start.
• Rate (R): the percentage charged or paid each year.
• Time (T or n): the number of years.
• Interest (I): the extra money earned or charged.
• Amount (A): the sum of the principal and the interest, A = P + I.

What is simple interest?

Simple interest is interest calculated on the principal only.

• Interest formula: I = PRT/100.
• Amount formula: A = P + PRT/100 = P(1 + RT/100).

Example: Find the amount on ₦360 borrowed for 5 1/2 years at 7% simple interest.

Solution:

Substituting the values:
A = 360(1 + (7 × 11)/(100 × 2))
= 360(1 + 0.385)
= 360 × 1.385
= ₦498.60

What is compound interest?

Compound interest is interest calculated on the amount at the end of each period, so the interest itself earns interest.

• Formula: A = P(1 + R/100)ⁿ, where n is the number of years.

How do simple and compound interest differ?

Example: Find the amount that ₦10,000 becomes if saved for 3 years at 8% per annum compound interest.

Method 1: year-by-year table

Amount = 11,664.00 + 933.12 = ₦12,597.12

Method 2: the formula

A = 10000(1 + 8/100)³ = 10000(1.08)³

1.08² = 1.1664, so 1.08³ = 1.1664 × 1.08 = 1.259712
A = 10000 × 1.259712 = ₦12,597.12

• Compound interest earned: 12,597.12 – 10,000 = ₦2,597.12.

Practice questions

1. What is simple interest?
2. Define compound interest.
3. Calculate 13 1/2% of ₦84,000.

Mixed practice questions

Choose the correct option.

1. Esther is 3 times as old as her sister Tolu. If the sum of their ages is 20 years, find the difference between their ages. (a) 20 years (b) 8 years (c) 10 years
2. 9 was subtracted from a certain number and the result was divided by 4. If the final answer is 5, what was the original number? (a) 29 (b) 18 (c) 20
3. A woman is 4 times as old as her son. In five years she will be 3 times as old as her son. How old is the woman? (a) 50 years (b) 40 years (c) 45 years
4. Bayo is 4 times as old as his sister Tolu. If the sum of their ages is 20 years, find the difference between their ages. (a) 12 years (b) 15 years (c) 18 years
5. Subtract the square root of 4 from the square of 4 and divide the result by 2. (a) 2 (b) 4 (c) 7

Evaluation

1. Solve x/5 + x/2 = 14.
2. Six workers build a wall in 12 days. How long will 9 workers take to build the same wall?
3. Find the amount on ₦5,000 saved for 2 years at 10% per annum compound interest.
4. Write the three consecutive even numbers whose sum is 48.

Assignment

Solve the following in your exercise book.

1. Divide 36 by the difference between the product of 3 and 6 and the square root of 36.
2. When I add 45 to a certain number and divide the sum by 2, the result is the same as five times the number. What is the number?

Further reading

• New Gen Maths for J.S.S. 3, pages 20 to 24
• Essential Mathematics for J.S.S. 3, pages 85 to 87