Percentage : The term **percent** means ‘for every hundred’. It can finest be clear as: “A portion whose denominator is 100 is called a percentage, and the numerator of the fraction is called the rate percent.”

**The following examples illustrate the percent’s and their fractional values:**

1) A student gets 60 percent marks in Arithmetic means that he obtained 60 marks out of every hundred of full marks. That is uncertainty the complete marks be 500, he gets 60 + 60 + 60 + 60 + 60 = 300 marks in mathematics. The overhead five 60s are one 60 for every hundred.

The total marks obtained by the student can be calculated in other ways, like, 60% OF 500 = 60/100×500 = 300

The beyond calculations can be finished easier by reducing the fractional value to its prime. As in the above case;

60% =60/100 = 3/5

If we remember that 60% = ^{3/5 }our calculation becomes easier. In

that case, the total marks obtained by the student = 3 /5_{ }x 500 = 300

2) A man invests 5% of his income into shares. It means:

- i) he invests 5 out of every 100 of his income into shares.
- ii) he invests — 5/100 of his income into shares.

iii) he invests —_{ }1/20th of his income into shares.

Now, if his income is 1050, how does he invest in shares?

Your quick answer should be 1050/20 = 52.5

We suggest you not to move with the fraction contain 100, if possible.

**Ex.1: The population of a town has increased from 60,000 to 65,000. Find the increase percent.**

**Soln:** Upsurge in population = 65, 000 — 60,000 = 5000

Percentage increase =5000/60000 x100=25/3=8 1/3%

**Ex.2: Ram’s salary is increased from Z630 to Z700. Find the increase per cent.**

**Soln:** Increase in salary =X700 —X630 =X70 Percentage increase

= 70/630×100 =111/9%

**Ex.3: In an election of two candidates, the candidate who gets 41% is rejected by a majority of 2412 votes. Find the total no. of votes polled.**

**Soln:** (59% — 41% 18% 2412 100% = ^{2412/18 }x100 =13400

**Ex.4: If 2 liters of water is evaporated on boiling from 8 liters of sugar explanation comprising 5% sugar, find the percentage of sugar in the remaining solution.**

**Soln:** As sugar has not been evaporated from the solution, the quantity of sugar in the original 8 liters of solution = the quantity of sugar in the Remaining 8 — 2 = 6 liters of solution i.e., 5% of 8 = x% of 6

**Second Method: % **of sugar in the original solution

= 5% of 8 liters = 0.4 liters

After evaporation of 2 **It **of water, the quantity of the remaining solution

= 8 — 2 = 6 liters

The required percentage of sugar = 0.4/6×100% = 6 2/3%

**Ex.5: Due to fall in manpower, the production in a factory decreases by 25%. By whatever percent must the in work hour be amplified to restore the original production?**

**Soln:** Decrease in production is only due to a decrease in manpower. Hence, manpower is decreased by 25%.

Now, presume that to reinstate the same production, working hours are increased by x%.

Making = Manpower x Working hours = M x W (say)

Now, M x W = (M 25% of M) x (W x% of W)

** **Or M×W = 75/100_{M x }100+x/100_{W}

_{or 100×100=75(100+x)}

or 400/3=100+x

**Ex.6: Express the fraction which 1.25 is of 10 as a percentage.**

**Soln:** The fraction = 1.25/10 = 125/1000 = 1/8

Now, 1/8 = 1/8 × 100/100 = 12 1/2/100 = 12 ½%

Note: The above question is often put as “what rate per cent is 1.25 of 10?

**Ex.7: What rate percent is 6P of Re 1?**

The fraction = 6p/1 = 6/100 = 3/50 = 3/50×100/100=6%

**Ex8: Two numbers are respectively 20% and 50% more than the third. What percentage is the first of the second?**

**Soln:** Subsequent the above statement, we have the required value

= 120/150×100 = 80%

Theorem: If A is x% of C and B is y% of C, then A is – x 100% of B.

**Ex.9: Two numbers are respectively 20% and 25% of a third number What percentage is the first of the second?**

**Soln:** Following the above theorem, we have the required value 20/25×100 = 80%

Note: The above relationships are very simple. When “What is the first of second” is asked, but the first as the numerator and the second as the denominator and vice versa.

**Ex.10: Two numbers are individually 30% and 40% less than a third number. What percent is the 2nd of the first?**

**Soln:** At first, you should find the formula yourself. If you can’t treasure it, go through the succeeding remarks.

(1) Subsequently, the two figures are less than the third; and (2) we have to find the percent of the second with respect to the first, our formula should be: 100-4/100-30×100=60/70×100=85 5/7%

**Ex.11: A positive number is divided by 5 instead of being multiplied by 5. What % is the outcome of the obligatory correct value?**

**Soln:** Let the no. be 1, then the correct answer = 5

The incorrect answer that was obtained =1/5

The reqd. % = 1/5×5×100%=4%

**Ex.12: A positive no is by error multiplied by 5 instead of being divided by 5. Via what percent extra or less than the correct answer is the result obtained?**

**Soln:** Let the no. be 1, then the correct answer = 1/5

The improper answer that was obtained = 5

:: The result is more than the correct answer by 5-1/5=24/5

The reqd 24/5/1/5×100% = 2400%