A percentage is a way of expressing a number as a fraction of 100 (percent meaning "per hundred"). It is often denoted using the percent sign, "%", or the abbreviation "pct". Since a percent is an amount per 100, percent can be represented as fractions with a denominator of 100. For example, 25% means 25 per 100, 25/100, and 350% means 350 per 100, 350/100. • A percent can be represented as a decimal. The following relationship characterizes how percents and decimals interact. Percent Form / 100 = Decimal Form
For example: What is 2% represented as a decimal? Percent Form / 100 = Decimal Form: 2%/100=0.02
Percent change
General formula for percent increase or decrease, (percent change): Percent=(Change/Original)∗100
Shortcuts
>If we want to measure ‘x %’ of a number ‘y’, we simply multiply ‘y’ by ‘x %’.For example: 20% of 50 = (20/100) x 50 = 10.
>100% of a number is the number itself. For example: 100 % of 50 = (100/100) x 50 = 50
>To calculate what % of a number ‘x’ is another number ‘y’, we multiply (y/x) by 100 For example: 10 is what % of 50? Answer: (10/50) x 100 = 20.
>If a number ‘x’ increases by 20 %, the result is x + (20 % of x). For example: If 50 increases by 20 %, the result is (50 + 20 % of 50) = (50 + 10) = 60. We can also say if a number ‘y’ increases by 20 %, the result is y multiplied by 1.2. For example: If 50 increases by 20 %, the result is 50 x 1.2 = 60.As, {(50 x 1) + (20/100) x 50} = {(50 x 1) + (0.2 x 50)} = 50 x (1 + 0.2) = 50 x 1.2 = 60
>If a number ‘x’ decreases by 20 %, the result is x – (20 % of x). For example: If 50 decreases by 20%, the result is (50 – 20 % of 50) = (50 - 10) = 40.We can also say if a number ‘y’ decreases by 20 %, the result is y multiplied by 0.8.For example: If 50 decreases by 20 %, the result is 50 x 0.8 = 40.As, {(50 x 1) – (20/100) x 50} = {(50 x 1) – (0.2 x 50)} = 50 x (1 – 0.2) = 50 x 0.8 = 40
VIDEO ON PERCENTAGE
Example:
1. What is 0.5 percent of 6.5?
A. 0.0325 B. 0.013 C. 0.325 D. 1.30 E. 130.0
Answer: A
2. Neha’s salary is 125% of Mehereen’s salary. Catherine’s salary is 80% of Mehereen’s salary. The total of all three salaries is $61,000. What is Catherine’s salary?
A. $15,000 B. $16,000 C. $17,000
D. $18,000 E. $19,000
Answer: B
3. If 8% tax on a sale amounts to 96 paisa, what is the final price (including tax) of the item?
A. $1.20 B. $2.16 C. $6.36 D. $12.00 E. $12.96
Answer: E
4. Among the 100 students in a business school, 50% enrolled in a marketing course. Of the enrolled students, 90% took the final exam. Two-thirds of the students who took the final exam passed it. How many students failed in the final exam?
A. 5 B. 10 C. 15 D. 30 E. 45
Answer: C
5. The price of sugar has increased by 60%. In order to restore the original price, the new price must be reduced by:
A. 33.33% B. 35% C. 37.5% D. 60% E. 66.66%
Answer: C
6. Twenty Prizes were distributed to five percent of the contestants. No contestant got more than one prize. The number of contestants were-
A. 200 B. 400 C. 300 D. 160 E. None of these
Answer: B
7. If the sales tax on a camera priced at 400 is between 4 percent and 7 percent, then the final price including tax is-
A. 404 B. 400 C. 415 D. 422 E. 436
Answer: D
8. The annual revenue of an agency increased by 25% last year. This year, it increased by 20%. If the increase in the exports was 1 million Taka last year, then what is the increase (in million Taka) this year?
A. 0.78 B. 0.8 C. 1 D. 1.2 E. 1.25
Answer: C
9. A chemist was preparing a solution that should have included 35 milligrams of a chemical. If he actually used 36.4 milligrams, what was his percentage error?
A. 0.04% B. 0.05% C. 1.40% D. 3.85% E. 4.00%
Answer: E
10. The average score on a certain examination was 80. Oindrila, on the same examination, scored 84. What was her percent deviation from the average score?
A. 6.7% B. 5.5% C. 5.0% D. 4.3% E. 4.0%
Answer: C
11. The price of a laptop is 200% of a TV. If the price of the TV increases by 20% and the price of the laptop decreases by 20%, what percent of the TV’s original price is the new combined price of the TV and Laptop?
A. 200 B. 220 C. 280 D. 300 E. 310
12. Oindrila’s salary went down from 81,000 Taka to 72,000 Taka as her company faced some loss. By what % should her salary be increased to restore her original salary?
A. 20% B. 16% C. 9% D. 12.5% E. 11.11%
13. In a box, 40% balls are white and rest are black. 60% balls are made of plastic & rest made of rubber. 25% white balls are made of plastic and 20 black balls are made of rubber. How many black balls are made of plastic?
A. 100 B. 80 C. 60 D. 50 E. 25
14. The price of mangoes went down by 20%. As a result, you can now buy 2 more kilos of mango with 480 Taka than you could do before. What is the previous cost of per kilo of mango?
A. 50 B. 60 C. 48 D. 42 E. 36
15. The cost of a watch is 3900 Taka with tax. If the tax free cost of the watch is 3000 Taka, what is the rate of tax?
A. 10% B. 20% C. 30% D. 40% E. 50%
16. The price of oil increased by 30% in June. It then went down by 30% in July. If the price of oil was 100 Taka per liter before the increase in June, what was the price of the oil after the price went down in July?
A. 90 B. 91 C.95 D. 100 E. 105
17. The cost of a good increased by 25%. Rafid intends to spend only 10% more than she already does behind this good. By how much should she cut back her consumption?
A. 9% B. 12% C.12.5% D. 15% E. 20%
Zero Net Change
>If a number increases by X/Y, then to restore it back to it’s original value, we need to decrease it by X/(X+Y).
For example: If a number increases by 20%, we can say it increases by 20/100 = 1/5
Here, X = 1 and Y = 5.
So, to restore it back to it’s original price, we need to decrease the number by 1/(1+5) = 1/6
We know, 1/6 = 0.1666 = 16.67%
So, to restore it back to it’s original price, we need to decrease the number by 16.67%
>If a number decreases by X/Y, then to restore it back to it’s original value, we need to increase it by X/(Y-X).
For example: If a number decreases by 20%, we can say it decreases by 20/100 = 1/5
Here, X = 1 and Y = 5.
So, to restore it back to it’s original price, we need to increase the number by 1/(5-1) = 1/4
We know, 1/4 = 0.25 = 25%
So, to restore it back to it’s original price, we need to increase the number by 25%.
Interest
Some important terms:
>Principal: The initial sum of money that is kept in a bank or lent out to others. Interest rate is
calculated on principal.
>Interest: The additional money paid back to the lender along with the principal is known as interest.
>Interest Rate: The proportion of a loan that is charged as interest to the borrower, typically
expressed as an annual percentage of the loan outstanding.
>Time Period: The duration for which the loan is due and the interest is accumulated.
>Amount: The sum total of the Principal and the total interest at the end of the time period. So, Amount = Principal + Interest
There are two types of interests –
Simple Interest
Simple interest = principal * interest rate * time, where "principal" is the starting amount and "rate" is the interest rate at which the money grows per a given period of time (note: express the rate as a decimal in the formula). Time must be expressed in the same units used for a time in the Rate.
Example: If $15,000 is invested at 10% simple annual interest, how much interest is earned after 9 months?
Solution: $15,000*0.1*9/12 = $1125
VIDEO ON SIMPLE INTEREST
Example:
1. How much simple interest will $2000 earn in 18 months at an annual rate of 6%?
A. Tk. 90 B. 120 C. 140 D. 160 E. 180
2. A moneylender charged Tk. 25 as simple interest on a loan of Tk. 150 for 1/6 years. What was the rate of interest per annum?
A. 125 B. 50 C. 75 D. 25 E. 100
3. In how many years taka 1800 will become taka 2250 if the simple interest rate is 5% p.a.?
A. 3 B. 4 C. 5 D. 6 E. None
4. Sara deposits $4,000 at a bank at an interest rate of 4.5% per year. How much interest will she earn at the end of 3 years?
A. $ 650 B. $ 620 C. $ 500 D. $ 540 E. $ 580
5. Russel keeps 5000 Taka in a bank at a x% simple interest rate. After 5 years his money grows to 6500 Taka. What is x?
A. 5 B. 6 C. 7 D. 8 E. 9
6. Two banks offered interest rates of 5% and 7% respectively on savings account. Adrito deposited a total amount of Tk. 4000 in the banks & in one year his interest income was Tk. 250. Find the investment in the bank with 7% interest.
A. 3000 B. 2000 C. 3500 D. 2500 E. None of these
7. A money lender lent Rs. 1000 at 3% per year and Rs. 1400 at 5% per year. The amount should be returned to him when the total interest comes to Rs. 350. Find the number of years.
A. 3.5 B. 3.75 C. 4 D. 4.5 E . 5
Compound Interest
Compound interest (or compounding interest) is interest calculated on the initial principal, which also includes all of the accumulated interest of previous periods of a deposit or loan. In compound interest, the principal amount with interest after the first unit of time becomes the principal for the next unit. The formula used to calculate standard compound interest (including the principal) is as follows:
A = P (1 + r)n
A is the final amount you repay at the end of the loan. P is the principal amount you borrow. r is the annual rate of interest. n is the number of periods you borrow/invest over.
Example:
1. Nisha puts Tk. 100 in the bank for two years at 5% interest compounded annually. At the end of the
two years, what will be her balance?
A. Tk. 100.00 B. Tk. 105.00 C. Tk. 105.25 D. Tk. 110.00 E. Tk. 110.25
2. A sum of money was put into bank. After 1 year the money grew to $330. If the rate of interest was
10% compounded annually, what was the initial amount put into the bank?
A. $250 B. $280 C. $300 D. $310 E. None of these
3. An investment earns 10% compounded annually. Find the value of an initial investment of $ 5,000 after 2 years.
A. $ 6,000 B. $ 6,050 C. $ 5,500
D. $ 5,050 E. None of these
4. A sum of money placed at compound interest doubles itself in 4 years. In how many years will it amount to 16 times itself?
A. 8 B. 12 C. 16 D. 20 E. 32
5. An amount of money grows to 3000 Taka in 3 years and 4000 Taka in 4 years on compound interest. What was the rate of interest?
A. 25% B. 33.33% C. 18% D. 20% E. Can’t be determined
Percentile
If someone's grade is in xth percentile of the n grades, this means that x of people out of n has the grades less than this person.
Example: Lena’s grade was in the 80th percentile out of 120 grades in her class. In another class of 200 students, there were 24 grades higher than Lena’s. If nobody had Lena’s grade, then Lena was what percentile of the two classes combined? Solution:
Being the 80th percentile out of 120 grades means Lena outscored 120∗0.8=96 classmates.
In another class, she would outscore 200−24=176 students. So, in combined classes, she outscored 96+176=272. As there are a total of 120+200=320 students, Lena is in 272/320=0.85=85, or in the 85th percentile.
Successive Change
If the value of something changes by a% and then undergoes b% change again, the net change can be calculated through the formula
±a±b±(ab/100)
use + when we are increasing the percentage/value, and use - when we are decreasing the percentage/value. The sign before (ab/100) will depend on the individual signs of a and b used beforehand. If both a and b are positive or both are negative, the sign before (ab/100) will be positive [as multiplication of plus and plus is positive. So is minus and minus]. But if one of them is of a different sign than the other, then the sign before (ab/100) will be negative [as multiplication of plus and minus is negative]. Use this formula accordingly
Example: The price of oil increased by 20% in a month. It again decreased by 20% the next month. What is the net % change in the price of oil? Solution:Here, a = 20 And, b = -20
So, a + b + (ab/100) = 20 + (-20) + [{20(–20)}/100] = –4. So, net % change is 4% decrease.
Video on “Percentage Problems”:
Percentage and Fractions
It is important to keep the following values in mind:
| 1/2 = 0.5 = 50% | 1/3 = 0.333 = 33.33% | 1/4 = 0.25 = 25% |
| 1/5 = 0.20 = 20% | 1/6 = 0.166 = 16.67% | 1/8 = 0.125 = 12.5% |
| 1/10 = 0.10 = 10% | 2/3 = 0.666 = 66.66% | 3/4 = 0.75 = 75% |