Sum and difference
The sum of a set of numbers is the result when the numbers are added together.
Example
The sum of four consecutive numbers is 58. Find the numbers.
Let the numbers be n, n+ 1, n + 2, n + 3.
n + (n + 1) + (n + 2) + (n + 3) = 58
4n + 6 = 58
Subtract 6 from both sides.
4n = 58 – 6
4n = 52
Divide both sides by 4.
4n = 52/4
n = 13
The numbers are 13, 14, 15, 16
The difference between two numbers is the result of subtracting one from the other. It is usual to subtract the smaller number from the larger. This gives a positive difference.
Example
The difference between 8 and another number is 7. Find two possible values for the number.
Let the number be x.
- Assuming x > 8, then
x – 8 = 17
Add 8 to both sides
x = 17 + 8 = 25
- Assuming x < 8, then 8 – x = 17
Add x to both sides
8 = 17 + x
Subtract 17 from both sides
8 – 17 = x
X = -9
Thus, the number could be 25 or -9.
Product
The product of two or more numbers is the result when the numbers are multiplied together.
Example
Find the product of -6, 0.7 and 6(2/3).
Product = -6 X 0.7 X 6(2/3)
= -6 X 7/10 X 20/3
= -6 X 7 X 20/10 X 3
= -2 X 7 X 2
= -28
Example
The product of two numbers is 8 4/9 if one of the numbers is 1/4 , find the other.
Let the number x.
¼ X x = 8 4/9
Multiply both sides by 4.
x = 8 4/9 X 4
= 33 7/9
Combining products with sums and differences
Example
Find the positive difference between 31 and the product of 4 and 14
Product of 4 and 14 = 4 X 14
= 56
Difference between 31 and 56 = 56 – 31
= 25
Notice that the problem is to find the difference between 31 and a product. Therefore, find the product first. (4 X 14) – 31 is equivalent to ‘the positive difference between 31 and the product of 4 and 14’.
Expressions with fractions
Example
Find one-quarter of the positive difference between 29 and 11.
Required value = ¼ (29 – 11)
= ¼(18)
= 18/4
= 4 ½
Example
Divide 40 by the sum of 3 and 5.
Required value = 40/3 + 5
= 40/8
= 5
Notice that the dividing line of a fraction acts like a bracket on the expressions above or below the line. For example,
40/3 + 5 = 40/(3 + 5)
Always simplify the expressions above or below the line before dividing out the fraction.
Example
Find the product of 10 and 6, subtract 23; then divide the result by 4.
Required value = (10 X 6) – 24/4
= 60 – 24/4
= 36/4
= 9
From numbers to words
Change the following numerical expressions into word statements
- (2 + 7) – 3
- 5(9 -6)
Solution
- (2 + 7) is ‘the sum of 2 and 7’.
(2 + 7) – 3 is ‘the positive difference between the sum of 2 and 7 and the number 3’ or ‘from the sum of 2 and 7, subtract 3’.
- 5(9 -6) is ‘the product of 5 and (9 – 6)’.
But (9 -6) is ‘the positive difference between 9 and 6’.
5(9 -6) is ‘the product of 5 and the positive difference between 9 and 6’.
Problems involving equations
The product of a certain number and 5 is equal to twice the number subtracted from 20. Find the number.
Let the number be x.
The product of x and 5 is 5x
Twice x is subtracted from 20 is 20 – 2x
Thus, 5x = 20 – 2x
Add 2x to both sides.
7x = 20
Divide both sides by 7.
x = 20/7 = 2 6/7
The number is 2 6/7.
Example
The sum of 35 and a certain is number is divided by 4. The result is equal to double the number. Find the number.
Let the number be n.
the sum 35 and nis the 35 + n
the sum divide by 4 is 35 + n/4
double the number is 2n
thus, 35 + n/4 = 2n
Multiply both sides by 4.
35 + n = 8n
Subtract n from both sides.
35 = 7n
Divide both sides by 7.
5 = n
The number is 5.
EXERCISES
Lets see how much you’ve learnt, attach the following answers to the comment below
- Find the sum of 82 and 148.
- Find the positive difference between 19 and 8.
- Find the difference between 63 and the product of `10 and 5.
- Find the difference between 27 and the product of 8 and 9.
- The sum of 8 and a certain number is equal to the product of the number and 3. Find the number.
- Four times a certain number is equal to the number subtracted from 40. Find the number.
