Classwork Series and Exercises {Mathematics- JSS3}: Word Problems

 

Word Problems

Word problems are mathematical problems which are written in words. This means that to be able to solve the problems we must understand the words that are often used in word problems. Some of these words are

Sum:The result we get when we add numbers together e.g. The sum of 2 and 3 = 2 + 3 = 5

Difference:The result of subtracting one number from another. It can be positive or negative

Positive difference:This is when we subtract the smaller number from the bigger number.

Example:  The positive difference between 2 & 5 = 5 – 2 = + 3

Negative difference:We get this by subtracting the bigger number from the smaller number.

Example: The negative difference between 2 & 5 = 2 – 5 = – 3

Product:The result of multiplication e.g. Product of 4 and 6 = 4 x 6 = 24

Consecutive:Numbers that follow each other e.g.  21, 22, 23, 24 are consecutive numbers

Consecutive odd numbers:Odd numbers that follow each other e.g. 21, 23, 25, 27…

Consecutive even numbers: Even numbers that follow each other e.g. 22, 24, 26, 28…

NOTE: The difference between consecutive numbers is 1 but the difference between consecutive odd/consecutive even numbers is 2.

Sum and Difference

We have said earlier that sum of a set of numbers is the result of adding the numbers together and the difference is what we get when we subtract one number from another. Now let us take some examples to help our understanding of the topic.

Example 1: Find the sum of 23, 27 and 33

Solution: Sum of numbers = 23 + 27 + 33 = 83 

Example 2: Find the positive difference between – 7 and – 12

Solution: Positive difference = Higher – Lower

Since – 7 is the higher number, positive difference will be – 7 – (– 12) = – 7 + 12 = +5

Example 3: The sum of three consecutive numbers is 63. Find the numbers.

Solution: When we have word problems where one or more of the quantities is unknown, we represent that quantity with a letter

So let the numbers be “t”, (t + 1) and (t + 1+1 = t +2) from difference between consecutive numbers

Therefore t + t + 1 + t + 2 = 63; 3t + 3 = 63; 3t = 60; t = 20

So the numbers are 20, 21 and 22

Example 4: The difference between – 3 and a number is 8. Find the two possible values for the number.

Solution:

Let the number be y; we are not told if it is positive or negative difference so we solve for both

Positive difference = y – (– 3) = 8; y + 3 = 8; y = 5

Negative difference= – 3 – y = 8

If we gather like terms; – 3 – 8 = y; y = – 11 

TRY THESE:

1. The sum of four consecutive odd numbers is 80. Find the numbers [17, 19, 21 & 23]

2. The difference between 12.6 and a number is 5.4. Find the two possible values of the number

    [7.2, 18.0]

Product

When we multiply 2 or more numbers, the result obtained is called the product of the numbers.

Example 1: Find the product of 2, 3 and 9

Solution: Product of 2, 3 and 9 = 2 x 3 x 9 = 54

Example 2: Find the product of 3³/8 and 3⁵/9

Solution: To multiply mixed fractions, it is important to convert them to improper fraction

3³/8 = ²⁷/8 and 3⁵/9 =³²/9

Therefore Product of 3³/8 and 3⁵/9= ²⁷/8 x ³²/9 = 12 

Example 3: The product of two numbers is 54. If one of the numbers is 27, find the other

Solution: Let the number be K. K x 27 = 54 so K = 54/27 = 2

Example 4: Find the product of –2, –5 and +9

Solution: Product = (–2) x (–5) x (+9) = +90

The final sign is + because the two negative (–) signs cancel out themselves 

TRY THIS:

1. The product of three numbers is 0.084. If two of the numbers are 0.7 and 0.2, find the third number [ANS: 0.6]

2. Find the product of 12, 0.6 and 6¼. [ANS: 45] 

Combined Operations

When we have the combination of addition, subtraction and multiplication in a word problem, the best thing to do is to take the operations one after the other following the BODMAS order of arithmetic operations.

Example 1: Find the sum of the product of ⁵/9 and ³/5and the product of ⁵/9 and ³/20

Solution:  We want to find (⁵/9 x ³/5) + (⁵/9 x ³/20)

(⁵/9 x ³/5) = ¹⁵/45 = ¹/3while (⁵/9 x ³/20) = ¹⁵/180) = ¹/12

So (⁵/9 x ³/5) + (⁵/9 x ³/20) = ⅓ + ¹/12 = ⁵/12

Example 2: Find the product of the difference between 2 and 7 and the sum of 2 and 7.

Solution:  Product of (difference between 2 and 7) and (sum of 2 and 7)

= (7 – 2) x (7 + 2) = 5 x 9 = 45

TRY THESE:

1. Find the difference between the sum of 16 and 17 and the product of 8 and 4. [ANS: 1]

2. Find the sum of the product of 9 and 5 and the product of 10 and 6. [ANS: 105]

3. Find the product of 6 and the sum of 2½ and 4 ½ [ANS: 42]

Expressions with fractions

When we have expressions with fractions follow the simple steps below, it will help you

Step I:  Interpret and write out the expressions

Step II: Simplify expressions in brackets (or the sub-expressions) 

Example 1:  Divide 52 by the sum of 1 and the product of 5 and 6

Solution: 52 ÷ [1 + (5 x6)]

= 52 ÷ (1 + 30) = 52 ÷ 31 = 1²¹/31

Example 2: Subtract 16 from the product of 4 and 10; then divide the result by 6

Solution:  [(4 x 10) – 16] ÷ 6 = (40 – 16) ÷ 6

= 24 ÷ 6 = 4

NOTE: The use of brackets is to make our work clearer and to make the solutions neater and more distinct. It also shows that we understand our expressions and we can interpret them correctly.

TRY THESE:

1. Divide 48 by the sum of 4 and 8 [ANS: 4]

2. Find one-seventh of the sum of 19 and the product of 4 and 11. [ANS: 9]

Expressions with equations

When we want to solve word problems that have equations it is important to do the following.

Step I: Write out the equations

Step II: Use LCM to clear out the fractions (if fractions are involved)

Step III: Carefully carry out the actions and take note of the negative sign if present. 

Example 1: 13 was subtracted from the product of 4 and a certain number. The result is equal to the sum of 5 and the original number. Find the number

Solution: Let the unknown number be a.

So (a x 4) – 13 = a + 5 ====> 4a – 13 = a + 5

Gather like terms; 4a – a = 5 + 13

3a = 18; a = 6

Example 2: Find the number that when ¾ of it is added to 3½, the sum is the same as when 2/3 of it is added to 6½

Solution: Let the number be f.

¾f + 3½ = 6½ + ⅔f

Convert the mixed fractions to improper fractions so that we can use LCM

¾f + ⁷/2 = ¹³/2 + ⅔f; LCM of all denominators is 12 so we would multiply though by 12

9f + 42 = 78 + 8f ====> 9f – 8f = 78 – 42; f = 36

TRY THIS:

1. I subtract 14 from a certain number. I multiply the result by. The final answer is 3. What was the number? [ANS: 15]

2. I add 12 to a number and then double their sum. The result is one and a half times what I get when I double the original number and add 12. Find the number. [ANS: 6]

Tests and Exercises

1. The sum of the square roots of 9 and 25 is… (a) 15 (b) 5 (c) 3 (d) 8

Guideline: Square root of 9 = 3; Square root of 25 = 5

Sum of the square roots = 3 + 5 = 8 

2. Find the number which when divided by 0.7 gives 0.4 (a) 0.28 (b) 1.28 (c) 1.18 (d) 28

Guideline: Let the number be y

Therefore, y ÷ 0.7 = 0.4; multiply both sides by 0.7

0.7 x (^y/0.7) = 0.4 x 0.7; y = 0.28

3. If a certain number is doubled, then the result divided by 7, the final result is 2. What is the number? (a) 5 (b) 6 (c) 7 (d) 8

Guideline: Let the number be P

(P x 2) ÷ 7 = 2; ^2P/7 = 2

Multiply both sides by 7====> 2P = 14; then divide the result by 2

P = 7 

4. Find one-sixth of the positive difference between 36 and 63 (a) – 4.5 (b) +3.6 (c) + 4.5 (d) +7

Guideline: Positive difference between 36 & 63 = 63 – 36 = 27

One-sixth of the difference = 27 ÷ 6 = +4.5

5. The sum of three consecutive even numbers is 72. The highest of the three numbers is (a) 18

    (b) 22 (c) 24 (d) 26

Guideline: Let the smallest number by m, so the next numbers will be (m+2) and (m+2+2) = (m + 4)

Therefore m + m+2 + m +4 = 72

3m + 6 = 80; 3m = 72 – 6; 3m = 66; m = 66/3 = 22

If m = 22, then the biggest number m + 4 = 22 + 4 = 26

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