## Arithmetic

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# Decimals, Fractions and Percentages

## Introduction & Example Values

Here is a table of commonly occuring values shown in Percent, Decimal and Fraction form:

 Percent Decimal Fraction 1% 0.01 1/100 5% 0.05 1/20 10% 0.1 1/10 12½% 0.125 1/8 20% 0.2 1/5 25% 0.25 1/4 331/3% 0.333… 1/3 50% 0.5 1/2 75% 0.75 3/4 80% 0.8 4/5 90% 0.9 9/10 99% 0.99 99/100 100% 1 125% 1.25 5/4 150% 1.5 3/2 200% 2

## Converting Between Percentage and Decimal

Percentage means “per 100”, so 50% means 50 per 100, or simply 50/100. If you divide 50 by 100 you get 0.5 (a decimal number). So:

• To convert from percentage to decimal: divide by 100 (and remove the “%” sign).
• To convert from decimal to percentage: multiply by 100 (and add a “%” sign).

The easiest way to multiply (or divide) by 100 is to move the decimal point 2 places. So:

## Converting Between Fractions and Decimals

The easiest way to convert a fraction to a decimal is to divide the top number by the bottom number (divide the numerator by the denominator in mathematical language)

### Example: Convert 2/5 to a decimal

Divide 2 by 5: 2 ÷ 5 = 0.4

Answer: 2/5 = 0.4

To convert a decimal to a fraction needs a little more work:

## Percentages

The terms percent means “for every hundred”. Percentage is a way of expressing a number as a fraction of 100. It is denoted by using the percent sign, %. For example, when we say a man made a profit of 20 percent we mean to say that he gained Rs. 20 for every hundred rupees he invested in the business.

Example: 25% = 25/100

What is 200% of 30?

200% × 30 = (200 / 100) × 30 = 60.

What is 13% of 98?

13% × 98 = (13 / 100) × 98 = 12.74.

60% of all university students are male. There are 2400 male students. How many students are in the university?

2400 = 60% × X, therefore X = (2400 / (60 / 100)) = 4000.

There are 300 cats in the village, and 75 of them are black. What is the percentage of black cats in that village?

75 = X% × 300 = (X / 100) × 300, so X = (75 / 300) × 100 = 25, and therefore X% = 25%.

The number of students at the university increased to 4620, compared to last year’s 4125, an absolute increase of 495 students. What is the percent increase?

495 = X% × 4125 = (X / 100) × 4125, so X = (495 / 4125) × 100 = 12, and therefore X% = 12%.

### Constant Product Rule

You can apply this rule when you have two parameters whose product is constant. In other words, when two parameters are inversely proportional to each other. For examples,

• Time × Speed = Distance
• Price × Consumption = Expenditure
• Length × Breadth = Area

The rule states that 1/x increase in one of the parameters will result in a 1/(x+1) decrease in the other parameter.

Let’s understand with the help of example. Suppose speed increases by 25% (or 1/4) and distance is constant, time required will decrease by 1/(4+1) or 1/5 or 20%.

### Percentage and Fraction Equivalents

If someone asks you to represent 50% in fractions then what will you do? Certainly, you will come with the answer 1/2. What this value actually represents? This is nothing but the fractional equivalent of the given percentage. From CAT point of view it is very important to know the fractional equivalent of the percentages.

### Multiplying Factor

While dealing with percentage increase or decrease, 10% increase is 1.1 and that of 15% decrease is 0.85.

• X increased by 10% would become X + 0.1 X = 1.1X
• X increased by 1% would become X + 0.01 X = 1.01X
• X decreased by 10% would become X – 0.1X = 0.9X
• X decreased by 1% would become X – 0.01 X = 0.99X
• X increased by 200% would become X + 2X = 3X
• X decreased by 300% would become X – 3X = −2X

### Successive Percentage Change

The population of a city increases by 10% in one year and again increases by 10% in the next year, then what is the net increase in the population in two years. The very common answer is 20% which is wrong. Why?

If Original population = P

After 1st year = 1.1P

After 2nd year = 1.21P

The population increases by 21% of the original value.

This successive change in the percentage can be calculated in the shortcut way as:

Consider a product of two quantities A = a x b.

If a & b change (increase or decrease) by a certain percentage say x & y respectively, then the overall percentage change in their product is given by the formula:

(x + y + xy/100)%

## Profit and Loss

Profit and Loss is mainly used in finance and business transactions.

Cost Price (CP): Expenses occured in producing a product or service.

Selling Price (SP): The price at which goods or services are sold.

Marked Price (MP): Price printed on the product for sale. If seller gives any discount, selling price will be different from the marked price.

### Profit and Loss Formulae

1. Profit (or Loss) = SP – CP (profit is made only when SP is greater then CP)
2. Profit % (or Loss %) = (Actual Profit/Loss ÷ CP) × 100%
3. CP = (SP × 100 ) ÷ (100 + profit %)
4. SP = (100 + profit %) × C.P ÷ 100
5. Actual Discount = MP – SP
6. Discount % = (Actual Discount ÷ MP) × 100%
7. SP = (MP – Discount %) of MP

## Interests

The lending and borrowing of money involves the concept of simple interest and compound interest. If you borrow money for certain period of time, you would have to return the this sum of money (Principal) with some extra money. this extra money is called Interest.

The money borrowed is called principal. The sum of interest and principal is called the amount. The time for which money is borrowed is called period.

Amount = Principal + Interest

The rate of interest is as per annum (unless indicated).

### Simple Interest

If the interest on a certain sum borrowed for a certain period is reckoned uniformly, then it is called Simple Interest and denoted as SI. Simple interest is simply calculated on principal amount using the following formula:

SI = (P × R × T)/100

where, P = principal, R = rate per annum, T = time in years

Amount can be calculated by adding interest to principal.

Example 1: A certain sum of money invested at some rate of interest triple itself in 4 years. In how many years the principal will become 9 times of itself at the same rate?

When the principal is in simple interest the interest for every year will be same. In 3 years the amount becomes 3 times the principal and we have

A = P + I

3P = P + I

⇒ I = 2P

The interest is 2 time the principal in 4 years or equal to principal in 2 years.

The interest will be equal to P in 2 years. So interest will be 8P in 16 years.

Amount after 16 years = P + 8P = 9P.

Hence the required answer will be 16 years.

### Compound Interest

When the borrower and the lender agree to fix up a certain unit of time (say yearly or half-yearly or quarterly) to settle the previous account. In such cases, the amount after the first unit of time becomes the principal for the second unit. The amount after second unit becomes the principal for the third unit and so on. After a specified period, the difference between the amount and the money borrowed is called Compound Interest for that period.

In case of compound interest, the total interest received in the present year will be added to the original principal and for the following year the principal will be Amount received (Principal + interest).

Suppose you lend Rs.10000 (principal) for 3 years at 10% per annum. So, you will get Rs.1000 as interest per annum (simple interest) For three years, interest will be Rs.3000 and thus you will get total amount of Rs.13000.

Compound interest involves interest on interest too, thus will give you better amount after 3 years. While calculating the compound interest, the principal amount keeps changing year after year (if the interest is compounded annually).

After 1 year: Interest = Rs.1000; New Principal = Rs.11000

After 2 years: Interest = Rs.1100; New Principal = Rs.12100

After 3 years: Interest = Rs.1210; You get Rs.13310. So, there is gain of Rs.310 if you lend at compound interest.

A = P(1 + r/n)nt

Example 2: A certain sum of money doubles in 3 years, then in how many years it will become 8 times at compound interest.

Think logically that in every 2 years, the principal becomes doubles of itself. So in 4 years it will be 4 times and in next two years it will be double of 4 times that is 8 times of original principal. So the required answer would be 8 years.

Example 3: A man took Rs. 5000 at 10% simple interest and gave it to another person at 10% compound interest, which is being compounded annually. After 3 years, how much extra money he will get?

Simple Interest on Rs.5000 = (5000 × 10 × 3)/100 = Rs. 1,500

He has to pay amount = Rs.5000 + Rs.1500 = Rs. 6500

Amount with compound interest = 5000(1 + 10/100)3 = Rs.6655

So, the answer is 6655 – 6500 = Rs 155.

SET THEORY

In some numerical problems, it is advantageous to use Venn diagrams, which are a powerful way to graphically organize information.

Sets A and B

The orange circle (set A) might represent, for example, all living creatures which are two-legged. The blue circle, (set B) might represent living creatures which can fly. The area where the blue and orange circles overlap (which is called the intersection) contains all living creatures which both can fly and which have two legs — for example, parrots.

The above logic can be used to solve lot of Logical Reasoning problems

Q7) There are 50 students admitted to a  class. Some students can speak only English and some can speak only Hindi. Ten students can speak both English and Hindi. If the number of students who can speak English is 21, then how many students can speak Hindi, how many can speak only Hindi and how many can speak only English?

1. a) 39, 29 and 11 respectively
2. b) 37, 29 and 13 respectively
3. c)  28, 18 and 22 respectively
4. d)  39, 11 and 29 respectively

A7) Correct answer is (a). Look at the venn diagram below. 29 students only speak Hindi, 10 speak both, so 21-10= 11 speak only English. Hindi speaking = 29+10=39

Q8) Out of a total of 120 musicians in a club, 5% can play all the three instruments – guitar, violin and flute. It so happens that the number of musicians who can play any two and only two of the above instruments is 30. The number of musicians who can play the guitar alone is 40. What is the total number of those who can play violin alone or flute alone?

1. a) 30
2. b) 38
3. c) 44
4. d) 45

A8) Correct answer is (c).

Let circles, G, V and F represent the musicians who can play guitar, violin and flute respectively. Now, a + b + c + d + e + f + g = 120. Number of musicians who can play all the three instruments = g = 5% of 120 = 6. Number of musicians who can play any two and only two of the instruments = d + e + f = 30. Number of musicians who can play guitar only = a = 40. Therefore, number of musicians who can play violin alone or flute only = 120 – (40+30+6) = 44.

Q9) A multinational firm hired a market research company to conduct a survey in the big metros re people breakfast eating habits. They were trying to target that segment of the population that is in a hurry in the morning and doesn’t have toast, eggs or cereal for breakfast. The results showed that 74% people have toast for breakfast, 42% have eggs and 52% have cereal. 36% have both cereal and egg, 52% have cereal and toast and 24% have toast and egg. 15% have all three. What percentage of population should the MNC target ?

1. a) 21% b) 15 %
2. c) 31% d) 43 %

Rather straight application of the set theory formula or you can do graphically also as in previous example.

A U B U C = A + B + C- (A U B + B U C + C U A) + A ∩ B ∩ C ,or in this case

74 + 42 + 52 – (36 + 52 + 24) + 15 , or 69. Therefore 100 – 69 = 31 percent do not usre any of the three products

Partnership , Ratio , Proportion & Variation

1. Partnership: When two or more than two persons run a business jointly, they     are called partners and the deal is known as partnership.

P(A) –🡪 T(A)

P(A)-🡪 I(A)

P(A) 🡪 T(A)X I(A)

P(B) 🡪T(B) X I(B)

P(A)        T(A)X I(A)

——  = ————

P(B)       T(B) X I(B)

1. Ratio of Division of Gains:

Suppose A and B invest Rs. x and Rs. y respectively for a year in a business, then at the end of the year:

(A’s share of profit) : (B’s share of profit) = x : y.

1. ii) When investments are for different time periods,

Suppose A invests Rs. x for p months and B invests Rs. y for q months, then

(A’s share of profit) : (B’s share of profit) = xp : yq.

` SOLVED EXAMPLES`

1.A, Band C start a business each investing Rs. 20,000. After 5 months A  withdrew Rs.6000 B withdrew Rs. 4000 and C invests Rs. 6000 more. At the end of the year, a total profit of Rs. 69,900 was recorded. Find the share of each

Sol. Ratio of the capitals of A, Band C

= 20000 x 5 + 15000 x 7 : 20000 x 5 + 16000 x 7 : 20000 x 5 + 26000 x 7

= 205000:212000 : 282000 = 205 : 212 : 282.

A’s share = Rs. 69900 x (205/699) = Rs. 20500    I

B’s share = Rs. 69900 x (212/699) = Rs. 21200;

C’s share = Rs. 69900 x (282/699) = Rs. 28200.

1. A, Band C enter into partnership. A invests 3 times as much as B

and B invests two-third of what C invests. At the end of the year, the profit earned is Rs. 6600. What is the share of B ?

Sol. Let C’s capital = Rs. x. Then, B’s capital = Rs. (2/3)x

A’s capital = Rs. (3 x (2/3).x) = Rs. 2x.

Ratio of their capitals = 2x : (2/3)x 😡 = 6 : 2 : 3.

Hence, B’s share = Rs. ( 6600 x (2/11))= Rs. 1200.

3.Four milkmen rented a pasture. A grazed 24 cows for 3 months; B 10 for  5 months; C 35 cows for 4 months and D 21 cows for 3 months. If A’s share of rent is Rs. 720, find the total rent of the field.

Sol. Ratio of shares of A, B, C, D = (24 x 3) : (10 x 5) : (35 x 4) : (21 x 3)     = 72 : 50 : 140 : 63.

Let total rent be Rs. x. Then, A’s share = Rs. (72x)/325

(72x)/325=720  x=(720 x 325)/72 = 3250

Hence, total rent of the field is Rs. 3250.

RATIO AND PROPORTION

1. RATIO: The ratio of two quantities a and b in the same units, is the fraction a/b and we write it as a:b.

In the ratio a:b, we call a as the first term or antecedent and b, the second  term or consequent.

## Ratio and Proportion

I When A : B = m : n, and B : C = p : q. Then A : B : C = mp : pn :nq.

Such questions are also widely asked in examinations like CLAT, so make sure that you don’t mug up such rules but your mind should absorb and internalize such tricks.

II When A : B = m : n, B : C = p : q, and C : D = r : s. Then A : B : C : D = mpr: npr : nqr : nqs.

Try to find a particular pattern being followed in both the tricks so that you don’t haveto memorize the equation.

Explanation: i) A : B = 1 : 2

1. ii) B : C = 3 : 4

iii) C : D = 5 : 6

Then A : B : C : D = 1*3*5 : 2*3*5 : 2*4*5 : 2*4*6

And, A : B : C : D : = 15 : 30 : 40 : 48

### Two-term Ratios

A ratio is a comparison between two quantities of the same kind, for example:

There are 3 red sweets and 5 yellow sweets in the box. We can say the ratio of red sweets to yellow sweets is 3 to 5. Ratio can be written with the symbol ‘:’ or as a fraction.

‘3 to 5’ can be written as ‘3:5’ or  When writing a ratio,

• change the quantities to the same unit if necessary
• reduce the ratio to its simplest form.

For example: What is the ratio of 5 minutes to 5 hours? First change the hours to minutes. 5 hours = 300 minutes Ratio = 5:300 = 1:60

### Three-term Ratios

A three-term ratio can be used to compare three quantities, for example:

There are 5 red sweets, 15 yellow sweets and 30 blue sweets in the box

5 to 15 to 30 = 5:15:30 =

Sometimes, you may need to convert 2 two-term ratios into 1 three-term ratio, for example:

If the ratio of the number of red shirts to the number of blue shirts is 1:2 and the ratio of blue shirts to green shirts is 1:3. What is the ratio of red shirts to green shirts?

First, you need to make the common item (in this case blue shirts) the same for both ratios. Convert the ratio of blue shirts to green shirts to its equivalent:

Next, combine to form three-term ratio: 1:2:6

1. (i) COMPARISON OF RATIOS:

We say that (a: b) > (c: d) <=>  (a/b)>(c /d).

(ii) COMPOUNDED RATIO:

The compounded ratio of the ratios (a: b), (c: d), (e : f) is (ace: bdf)

1. (i) Duplicate ratio of (a : b) is (a2 : b2).

(ii) Sub-duplicate ratio of (a : b) is (a : √b).

(iii)Triplicate ratio of (a : b) is (a3 : b3).

(iv) Sub-triplicate ratio of (a : b) is (a ⅓ : b ⅓ ).

(v) If (a/b)=(c/d), then  ((a+b)/(a-b))=((c+d)/(c-d))

(Componendo and dividendo)

Proportions/Variations

PROPORTION: The equality of two ratios is called proportion.

If a: b = c: d, we write, a: b:: c : d and we say that a, b, c, d are in proportion . Here a and d are called extremes, while b and c are called mean terms.

Product of means = Product of extremes.

Thus, a: b:: c : d <=> (b x c) = (a x d).

1. (i) Fourth Proportional: If a : b = c: d, then d is called the fourth proportional

to a, b, c.

(ii) Third Proportional: If a: b = b: c, then c is called the third proportional to

a and b.

(iii) Mean Proportional: Mean proportional between a and b is square root of ab

Ex. Find:

(i) the fourth proportional to 4, 9, 12;

(ii) the third proportional to 16 and 36;

iii) the mean proportional between 0.08 and 0.18.

Sol.

1. i) Let the fourth proportional to 4, 9, 12 be x

Then, 4 : 9 : : 12 : x  ⬄4 x x=9×12 ⬄ X=(9 x 12)/14=27;

Fourth proportional to 4, 9, 12 is 27.

(ii) Let the third proportional to 16 and 36 be x.

Then, 16 : 36 : : 36 : x  ⬄16 x x = 36 x 36 ⬄ x=(36 x 36)/16 =81

Third proportional to 16 and 36 is 81.

(iii) Mean proportional between 0.08 and 0.18

√0.08 x 0.18 =√8/100 x 18/100= √144/(100 x 100)=12/100=0.12

1. VARIATION:

(i) We say that x is directly proportional to y, if x = ky  for some constant k and

we write, x ∝ y.

(ii) We say that x is inversely proportional to y, if xy = k for some constant k and

we write, x∞(1/y)

### Direct Proportions/Variations

Two values x and y are directly proportional to each other when the ratio x : y or is a constant (i.e. always remains the same). This would mean that x and y will either increase together or decrease together by an amount that would not change the ratio.

Knowing that the ratio does not change allows you to form an equation to find the value of an unknown variable, for example:

If two pencils cost \$1.50, how many pencils can you buy with \$9.00?

The number of pencils is directly proportional to the cost.

pencils

### Inverse Proportions/Variations

Two values x and y are inversely proportional to each other when their product xy is a constant (always remains the same). This means that when x increases y will decrease, and vice versa, by an amount such that xy remains the same.

Knowing that the product does not change also allows you to form an equation to find the value of an unknown variable for example:

It takes 4 men 6 hours to repair a road. How long will it take 8 men to do the job if they work at the same rate?

The number of men is inversely proportional to the time taken to do the job.

hours.

Usually, you will be able to decide from the question whether the values are directly proportional or inversely proportional.

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