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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