How to find percentage:
A percentage of any quantity is a degree by which that very quantity exists. It is another form of Proportion, Ratio or even Fraction.
Note that a percentage of any quantity at its normal existence is 100 and it is represented with the simbol “%”.
Examples of percentages are:
x%, y%, z% etc.
Those letters listed above can be any numbers, they can be either wholes, fractions or decimals.
Percentage as a fraction:
If for instance you see something like 5%, it is written as 5/100. 6% is written as 6/100. Generally if “x” is any whole number in percentage (x%), it is written as x/100 provided that x ≤ 100.
Example 1:
Caleb scored 45 out of 75 marks in a Test, what percentage is this?
Solution: Using method 1:
His score = 45
Total score = 75
Let “y%” be the percentage of the 45 marks scored by Caleb and 100% is for the 75 marks.
Thus,
45 = y
75 = 100
Crossing multiplying will give us 75y = 100 x 45
75y = 4500
Divide both sides by the coefficient of y.
y = 4500/75
y = 60%
Method 2:
He scored 45 marks
Out of 75 marks
The mark is being written normally as 45/75.
Multiply it by the percentage of the normal existence of the mark (75) which is 100%
So, 45/75 x 100
= 3/5 x 100
= 3 x 20 = 60%
Example 2:
Mr Gilbert spent 25% of his salary on transportation, 30% of the rest on rent and the balance left is #1,932. How much was spent on transportation?
Solution:
Let Mr Gilbert’s salary be “h”
He spent (25/100) x h on Transportation.
= (1/4) x h = h/4
Remaining amount = h – (h/4)
= (4h – h)/4
= 3h/4
He spent (30/100) x (3h/4) on rent
= (3/10) x (3h/4)
= 9h/40
Balance = (3h/4) – (9h/40)
= (30h – 9h)/40
= 21h/40 = 1932
Cross multiply.
21h = 40 x 1932
21h = 77280
Divide both sides by the coefficient of h.
h = 77280/21
h = #3,680
Therefore he spent 25% of his salary on transportation.
(25/100) x #3,680
= 92000/100
= #920
Percentage Increase or Decrease
A percentage Increase is a degree by which a quantity rises above its normal existence 100%. while a percentage decrease is a degree by which a quantity reduces below its normal existence 100%.
Example 1:
The price of an IntellectSolver’s mathematics textbook in Charity’s Bookshop is #5,000, she sold the textbook #5,235. What is the percentage Increase in the sell?
Solution:
Actual price = #5,000
Selling price = #5,235
Increase = 5,235 – 5000
= #235
Using first method:
Let the percentage Increase be “m”
235 = m
5000 = 100
Cross multiply.
5000m = 100 x 235
5000m = 23500
Divide both sides by the coefficient of m.
m = 23500/5000
m = 4.7%
So Charity made an extra 4.7% gain.
Example 2:
A POS Attendance earned a salary of #15,000 in 2020 and earned 20% more in 2021. If the Attendance always give her Mother 15%, find what she gave to her mother in 2021.
Solution:
In 2020, she earned #15,000
In 2021, she earned 20% more
= [(20/100) x 15000] + 15000 = [(1/5) x 15000] + 15000
= 3000 + 15000 = #18,000
The amount she gave her mother in 2021.
= (15/100) x 18,000
= #2,700
QUESTIONS AND SOLUTIONS:
Q1: In a school, 1200 of the students are boys. If 50% of the boys and 40% of the girls have paid their school fees, find the number of girls, given that 46% of the population has paid their school fees.
Solution:
Total number of boys = 1200
And 50% of the boys have paid their school fees:- (50/100) x 1200 = 600 boys.
Let total number of girls be “y”
And 40% of the girls have paid their school fees:- (40/100) x y = 2y/5 girls.
Total number of students that have paid their school fees
= 600 + (2y/5)
= (3000 + 2y)/5
Again, total number of the population that have paid their school fees
= (46/100) x (1200 + y)
= (55200 + 46y)/100
(3000 + 2y)/5 = (55200 + 46y)/100
Cross multiply.
100(3000 + 2y) = 5(55200 + 46y)
20(3000 + 2y) = 55200 + 46y
60000 + 40y = 55200 + 46y
Collection of like terms
40y – 46y = 55200 – 60000
-6y = -4800
Divide both sides by the coefficient of y.
y = -4800/-6
y = 800 girls.
Q2: A man’s wages increased from #6,250 to #7,125 per month.
(a) Find the increase percent.
(b) If the wages are taxed at 12%, find the increase in tax payable.
Solution:
Previous wages = #6,250
Current wages = #7,125
Increase in wages
= 7125 – 6250 = #825
(a) Increase percent
= (Increase/Actual) x 100 = (825/6250) x 100
= 82500/6250 = 13.5% = 14% (nearest whole number)
(b) Increase in tax payable
Tax he paid before the increase in wages
= (12/100) x 6250 = 75000/100
= #750
Tax he paid after the increase in wages
(12/100) x 7125
= 85500/100
= #855
Therefore, Increase in tax payable = 855 – 750
= #105
Q3: A worker’s period of duty increases from 56 hours to 72 hours but the wages decrease by 15%. By what percentage do the new total wages increase or decrease?
Solution:
Let W represents the original hourly wage.
The wages decrease by 15%, so the new percentage for the new wages = (100 – 15)% = 85%
Original total wage = 56 x W = 56W
Decrease hourly wage = (85/100) x W
Decrease total hourly wage = 72 x (85/100) x W
= 61.2W
Percentage Increase
= [(61.2W – 56W)/56W] x 100
= (5.2/56/ x 100 = 520/56 = 9.29 = 9.3%
Q4: Divide #3,420 into two parts, such that one part is 28% more than the other.
Solution:
Let one part be T.
The other part = T + [(28/100) x T]
= T + (28T/100) = (100T + 28T)/100 = 128T/100
The sum of the two parts (128T/100) + T = 3,420
128T + 100T = 342000
228T = 342000
Divide both sides by the coefficient of T.
T = 342000/228
T = #1,500
The other part
= T + (28T/100)
= 1500 + (28 x 1500)/100 = 1500 + (28 x 15)
= 1500 + 420 = #1,920
Q5: When the weight of a baby decreases by 16%, it becomes 10.5kg. Find what the weight would have been if it increased by 9%.
Solution:
Let the baby’s formal weight be Q.
But 16% of the weight
= (16/100) x Q
= 16Q/100
This baby’s new weight
= Q – (16Q/100) = (100Q – 16Q)/100
= 84Q/100
84Q/100 = 10.5kg
Cross multiply
84Q = 1050
Q = 1050/84
Q = 12.5kg
If the weight is increased by 9%
Weight Increase
= (9/100) x 12.5kg = 112.5/100= 1.125kg
Total weight now
= (12.5 + 1.125)kg = 13.625kg
= 13.6kg (1 d.p)
Q6: The diameter of a circle increased from 20m to 20.8m. Find the percentage Increase in its circumference.
Solution:
Circumference of a circle (C) = 2πr = πD
D1 = 20m
C1 = 3.142 x 20 = 62.84m
D2 = 20.8m
C2 = 3.242 x 20.8 = 65.3536m = 65.35m
The percentage Increase in the circumference
= [(65.35 – 62.84)/65.35] x 100
= (2.51/65.35) x 100 = 251/65.84
= 3.81% = 4% ( nearest whole number)
Q7: A hotelier thinks to welcome 565 guests but has overstated by 25% . How many guests does he actually welcome?
Solution:
Let the number of guests he welcomed be y.
Number overstated by 25% = (25/100) x y
= y/4
But y + y/4 = 565
4y + y = 2260
5y = 2260
y= 452 guests
Q8: A university estimated the number of candidates for admission in a new session. When the season began, there were 2800 candidates and the university realised an overstatement of 2½%. How many candidates were expected?
Solution:
Let the number of candidates the university estimated be “C”
Number overstated = (2½/100) x C = (5/200) x C = C/40
C/40 candidates.
Total number of candidates that came to the university
= C – (C/40) = 2800
40C – C = 112000
39C = 112000
C = 280000/97.5
C = 2871
Q9: Find the sum of increase by 24% of #720 and decrease by 18% of #430.00
Solution:
24% of #720
= (24/100) x 720
= #172.80
Increase = 720 + 172.80
= #892.80
18% of #430.00
= (18/100) x 430
= #77.40
Decrease = 430 – 77.4
= #352.60
The sum of the two
= 892.80 + 352.60
= #1245.40
Q10: A legacy of #4,500 was invested at 20%. The eldest son takes 50%, the second son 30% and the youngest takes 90% of the remaining. What percentage of the original legacy does the youngest receive?
Solution:
The eldest son takes 50% = (50/100) x 4500
= #2,250
The second son takes 30% = (30/100) x 4500
= #1,350
Remainder = #4500 – (#2,250 + #1,350)
= #4500 – #3,600
= #900
The youngest takes 90% of the remainder
= (90/100) x 900
= #810
Therefore the percentage of the original legacy the youngest receive
= (810/4500) x 100 = 810/45
= 18%
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