Word problems are seen in our everyday lives. This involves sum, difference, positive difference and product of numbers. The most important thing you need to remember in solving word problems is the INTERPRETATION OF THE QUESTION. If you’re able to correctly interpret the question, the solution becomes easy.
Sum – the result of addition
Difference – the result of subtraction
Positive difference – larger number minus smaller number
Product – the result of multiplication
The sum of a set of numbers is the result when the numbers are added together.
Example 1:
The sum of four consecutive numbers is 78.
Find the numbers.
Let the numbers be a, a+1, a+2, a+3.
a + (a+1) + (a+2) + (a+3) = 78
4a + 6 = 78
Subtract 6 from both sides
4a + 6 – 6 = 78 – 6
4a = 72
Divide both sides by 4
4a/4 = 72/4
a = 18.
The numbers 18, a+1 = 19, a+2 = 20, a+3 = 21
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 one. This gives a positive difference.
Example 2:
The difference between 7 and another number is 12. Find two possible values for the number.
Let the number be x
i. Assuming x > 7, then
x – 7 = 12
Add 7 to both sides
x – 7 + 7 = 12 + 7
x = 19
ii. Assuming x < 7, then
7 – x = 12
Add x to both sides
7 – x + x = 12 + x
7 = 12 + x
7 – 12 = x
x = -5
Thus the number could be 19 or -5.
The product of two numbers is the result when the numbers are multiplied together
Example 3:
Find the product of -6, 0.7 and 6 2/3.
Convert 0.7 to a proper fraction = 7/10, 6 2/3 = 20/3
Product = -6 x 7/10 x 20/3
= -6 x 7 x 20
10 x 3
= -2 x 7 x 2
= -28
Example 4:
The product of two numbers is 8 4/9. If one of the numbers is 1/4, find the other number.
Let the number be a.
1/4 x a = 8 4/9
Multiply both sides by 4
1/4 x a x 4 = 76/9 x 4
a = 33 7/9
Combining products with Sum and Differences
Example 5:
Find the positive difference between 45 and the product of 4 and 15
Product of 4 and 15 = 60
Difference between 45 and 60 = 60 – 45 = 15.
Examples 6:
Find the product of 8 and the positive difference between 3 and 9.
Positive difference = 9 – 3 = 6
Product = 8 x 6 = 48
Example 7:
Find the sum of 2.5 and the product of 3 and 2.5
Sum and Product = 2.5 + {3 x 2.5}
= 2.5 + {7.5}
= 10.0
Problems Involving Equations
Example 8:
The product of a certain number and 8 is equal to twice the number subtracted from 24. Find the number
Let the number be x
The product of x and 8 = 8x
twice x (2x) subtracted from 24 = 24 – 2x
thus, 8x = 24 – 2x
Add 2x to both sides
8x + 2x = 24 – 2x + 2x
10x = 24
Divide both sides by 10
10x/10 = 24/10
x = 2.4
Example 9:
The sum of 42and a certain number is divided by 4. The result is equal to double the number. Find the number
Let the number be d
the sum of 42 and the number = 42 + d
the sum divided by 4 = 42 + d
4
result is double the number 2d = 42+ d = 2d
4
multiply both sides by 4 = 42 + d = 2d x 4
42 + d = 8d
subtract d from both sides
42 + d – d = 8d – d
42 = 7d
divide both sides by 7
42/7 = 7d/7
d = 6
Example 10:
The sum of two numbers is 22. The sum of 3/4 of one of the numbers and 1/5 of the other number is 11. Find the two numbers.
Let the numbers be a and b = a + b = 22
The sum of 3/4 of one number (3/4 a) and 1/5 of the other number (1/5 b) is 11 = 3/4 a + 1/5 b = 11
Create a common variable
a + b = 22, therefore b = 22 – a
Substitute b into second equation
3/4 a + 1/5 (22 – a) = 11
Find the LCM
3 + 22 -a = 11
4 5
15a + 88 – 4a = 11
20
Cross multiply
15a + 88 – 4a = 20 x 11
15a – 4a = 220 – 88
11a = 132
Divide both sides by 11
a = 12.
a + b = 22
12 + b = 22
b = 22 – 12
b = 10.
Classwork Exercises
- 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
- I subtract 14 from a certain number, I multiply the result by 3. The final answer is 3. Find the number
- The sum of two numbers is 21. Five times the first number added to 2 times the second number is 66. Find the two numbers.
- 2 is added to twice a certain number and the sum is doubled. The result is 10 less than 5 times the original number. Find the original number.
- The sum of two numbers is 38. When 8 is added to twice one of the numbers, the result is 5 times the other number.
- 5/12 of a number is subtracted from 3/4 of the number. Their positive difference is 7 less than 5/6 of the number. Find the number.
- Find the number such that when 3/4 of it is added to 3 1/2, the sum is the same as when 2/3 of it is subtracted from 6 1/2.
- The sum of two numbers is 21. 3/4 of one of the numbers added to 2/3 of the other gives a sum of 15. Find the two numbers.
- 1/3 of a number is added to 5. The result is one and half times the original number. Find the number.
Solution
- The sum of 8 and a certain number is equal to the product of the number and 3. Find the number
Let the number be x.
The sum of 8 and x = 8 + x
The product of the number and 3 = 3x
Sum of 8 and x (8 + x) is equal to the product of the number and 3 (3x) = 8 + x = 3x
Subtract x from both sides of the equation = 8 + x – x = 3x -x
= 8 = 2x
Divide both sides by 2 = x = 4. - Four times a certain number is equal to the number subtracted from 40. Find the number
Let the number be a
Four times the number = 4 x a = 4a
four times the number (4a) is equal to the number subtracted from 40 (40 – a): 4a = 40 – a
Add a to both sides = 4a + a = 40 – a + a
= 5a = 40
Divide both sides by 5 = 5a/5 = 40/5
Therefore, a = 8 - I subtract 14 from a certain number, I multiply the result by 3. The final answer is 3. Find the number
Let the number be c
Subtract 14 from the number = c – 14
Let the result be d, so c – 14 = d
Multiply the result by 3 = 3 x d = 3d
Final result is 3, hence 3d =3
Divide both sides by 3, d = 1
Remember c – 14 = d (1)
c – 14 = 1
c = 1 + 14 = 15 - The sum of two numbers is 21. Five times the first number added to 2 times the second number is 66. Find the two numbers.
Let the 2 numbers be x and y
Sum of the 2 numbers = x + y = 21
Five times the first number (5x) is added to 2 times the second number (2y) to give 66= 5x + 2y = 66
There are 2 equations – x + y = 21 ……….. eqn 1
– 5x + 2y = 66 ……… eqn 2
Eliminate one variable by multiplying equation 1 with 2 and equation 2 with 1
– x + y = 21 x 2 = 2x + 2y = 42 …….. eqn 1
– 5x + 2y = 66 x 1 = 5x + 2y = 66 ……eqn 2
Subtract equation 1 from 2 = 5x +2y= 66 – 2x +2y= 42
New equation = 3x = 24
Divide through by 3, x = 8
To find y = Pick any of the equations (8) + y = 21, y = 21 – 8 = 13
Or 5(8) + 2y = 66, 2y = 66 – 40 = 26, 2y = 26, y = 13.
The two numbers are 8 and 13. - 2 is added to twice a certain number and the sum is doubled. The result is 10 less than 5 times the original number. Find the original number.
Let the number be z
2 is added to twice the number = (2 + 2z)
The sum is doubled = (2 + 2z) + (2 +2z)
Result is 10 less than the original number = 5z – 10
Complete equation is = (2 + 2z) + (2 +2z) = 5z – 10
= 2 + 2z + 2 + 2z = 4 + 4z
4 + 4z = 5z – 10
Collect like terms to either sides of the equation
4z – 5z = -10 – 4
-z = -14
Divide through by -, z = 14
6. The sum of two numbers is 38. When 8 is added to twice one of the numbers, the result is 5 times the other number.
Let the two numbers be x and y
Sum of x and y = x + y = 38
8 is added to twice a number = 8 + 2x , the result is 5 times the other number = 8 + 2x = 5y
Two equations x + y = 38 and 8 + 2x = 5y
Rearrange equation 2 = -2x + 5y = 8
Solve simultaneously x + y = 38 …….. eqn 1
-2x + 5y = 8 ……. eqn 2
Multiply equation 1 by 2 and equation 2 by 1 – x + y = 38 x 2 = 2x + 2y = 76
-2x + 5y = 8 x 1= -2x + 5y = 8
In order to eliminate one variable, add the two equations together =2x+ 2y = 76 …….. eqn 1
+(-2x)+ 5y = 8 ……. eqn 2
7y = 84, y = 12
To solve for x, take any of the equations above = x + (12) = 38, x = 38 -12 = 26
The two numbers are 26 and 12.Try and solve the rest of the exercise yourself. Follow the examples and solution patterns above.
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