RATIO PROPORTION & VARIATION

IMPORTANT RESULTS:

1.A ratio is an ordered comparison of two quantities. The ratio of a and b is denoted by a : b.

2.The equality of ratios is called proportion .If a : b = c : d then we say that a, b, c, d are in proportion.

3.If a : b = c : d then d is called the fourth proportion of a, b, c.

4. If a : b = b : c then c is called the third proportion of a, b, c.

5.Mean proportional between a and b is √ab.

6.Duplicate ratio of a : b is a2 : b2.

7.Sub – duplicate ratio of a : b is √a : √b.

8.Triplicate ratio of a : b is a3 : b3.

9.Sub- triplicate ratio of a : b is a1/3 : b1/3.

10.we say that x is directly proportional to y, if x = k y for some constant k .

11. we say that x is inversely proportional to y, if x y = k for some constant k .

PROBLEMS :

1.Find the ratio of third proportional to 12 and 30 and the mean proportional of 9 and 25?

2. In an examination, 25 students out of 70 scored less than 50% marks. Find the ratio of number of students who scored 50% marks or more to number of students who scored less than 50% marks?

3. A sum Rs.427 is to be divided among A,B and C such that 3 times A’s share , 4 times B’s share and 7 times C’s share are all equal. The share of C is

4. If 35% of A’s income is equal to 25% of B’s income, then the ratio of their income is

5. 5 mangoes and 4 oranges costs as much as 3 mangoes and 7 oranges. The ratio of the cost of one mango to that of one orange is

6. The ratio between two numbers is 3 : 4 and their L. C. M is 180. The first number is :

7. If a : b = b : c = 2 : 3 find a:b:c

8. Find a:e if a:b =2:3, b:c=6:7, c:d=14:25 and d:e =1:2

9. Two numbers are in the ratio of 2:5. If 4 is added to each, they would be in the ratio 4:9. Find the numbers

10. The ratio of the ages of A, B and C is 7 : 5 : 4. If C’s age is 32 years, find the sum of the ages of A,B and C

11. x varies inversely as square of y. Given that y = 2 for x = 1. The value of x for y = 6 will be equal

PERCENTAGES

Important Results:

I. Concept of Percentage: By a certain percent, we mean that , many hundredths. Thus , x percent means x hundredths, written as x%

To express x% as a fraction: we have, x% = x

100

Thus, 20% = 20

100 = 15 ; 48% =

40100 =

1225 , etc.

To express ab as a percent : We have ,

ab = (

ab

X 100)%

Thus, 14 = (

14

X 100)% = 25%; 0.6 = 6

10 = 35 = (

35 X 100)% = 60%

II. If the price of a commodity increases by R%, then the reduction in consumption so as not to increase the expenditure is

⦋ R(100+R)

X 100 %⦌

If the price of a commodity increases by R%, then the reduction in consumption so as not to increase the expenditure is

⦋ R(100−R)

X 100 %⦌

III. Results on Population: Let the population of a town be P now and suppose it increases at the rate of R% per annum, then:

1. Population after n years = P (1+ R100 ) ⁿ

2. Population n years ago = P

(1+ R100

)ⁿ

IV. Results on Depreciation : Let the present value of a machine be P. Suppose it depreciates at the rate of R% per annum. Then:

1. Value of the machine after n years = P (1− R100 ) ⁿ

2. Value of the machine r n years ago = P

(1− R100

) ⁿ

V. If A is R% more than B, then B is less than A by

⦋ R(100+R)

X 100 %⦌

If A is R% Less than B, then B is more than A by

⦋ R(100−R)

X 100 %⦌

PROBLEMS:

1.If A is 20% of C and B is 25% of C, then what percentage is A of B?

2. If A’s salary is 10% more than that of B, then how much percent is B’s salary less than that of A?

3. Two numbers are 32% and 20% less than a third number respectively. What percent is the first of the second?

4. In an examination, A Scored 25% less than B. By what percent did B score more than A

5. The population of a town increases by 5% annually. If its population in 2001 was 1,38,915, what it was in 1998

6.Sixty five percent of a number is 21 less than four-fifth of that number. What is the number.

7.Diffrence of two numbers is 1660. If 7.5% of one number is 12.5% of the other number, find two numbers.

8. In an election between two candidates, 75% of the voters cast their votes, out of which 2% of the votes were declared invalid. A candidate got 9361 votes which were 75% of the total valid votes. Find the total number of votes enrolled in that election.

9.If the numerator of a fraction be increased by 15% and its denominator be diminished by 8%, the value of the fraction is 15 / 16. Find the original fraction.

10. In ann examination, 35% of total students failed in Hindi, 45% failed in English and 20% in both. Find the percentage of those who passed in both the subjects.

PROFIT AND LOSS

Important Results:

Cost Price: The price at which an article is purchased, is called its Cost Price, abbreviated as C.P.

Selling Price: The price at which an article is sold, is called its Selling Price, abbreviated as S.P.

Profit or Gain: If S.P. is greater than C.P., the seller is said to have a Profit or Gain.

Loss: If S.P. is less than C.P., the seller is said to have incurred a loss.

1. Gain = (S.P.) – (C.P.)

2. Loss = (C.P.) – (S.P.)

3.Loss or gain is always reckoned on C.P.

4. Gain % = (Gain x 100

C . P . )

5.Loss % = (Loss x100

C . P . )

6. S.P. = (100+Gain %

100 ) x C.P.

7. S.P. = (100−Loss %

100 ) x C.P.

8. C.P. = (100

100+Gain % ) x S.P.

9. C.P. = (100

100−Loss % ) x S.P.

10. If an article is sold at a gain of says, 35 %, then S.P. = 135% of C.P.

11. If an article is sold at a loss of says, 35 %, then S.P. = 65% of C.P.

12. When a person sells two similar items, one at a gain of says , x% , and the other at a loss of x% , then the seller always incurs a los given by:

Loss % = (Commom Loss∧Gain %

10 )² = (X10 )²

13. If a trader professes to sell his goods at cost price, but uses false weights, then

Gain % = (Error %

(TrueValue )−(Error)x100)%

Problems:

1.Alfred buys an old scooter for Rs. 4700 and spends Rs. 800 on its repairs. If he sells the scooter for Rs. 5800, his gain percent is.

2.The cost price of 20 articles is the same as the selling price of x articles. If the profit is 25%, then the value of x is:

3.A vendor bought toffees at 6 for a rupee. How many for a rupee must he sell to gain 20%?

4.A man buys a cycle for Rs. 1400 and sells it at a loss of 15%. What is the selling price of the cycle?

5.Sam purchased 20 dozens of toys at the rate of Rs. 375 per dozen. He sold each one of them at the rate of Rs. 33. What was his percentage profit?

6.Some articles were bought at 6 articles for Rs. 5 and sold at 5 articles for Rs. 6. Gain percent is:

7.When a plot is sold for Rs. 18,700, the owner loses 15%. At what price must that plot be sold in order to gain 15%

8.100 oranges are bought at the rate of Rs. 350 and sold at the rate of Rs. 48 per dozen. The percentage of profit or loss is:

9.A shopkeeper sells one transistor for Rs. 840 at a gain of 20% and another for Rs. 960 at a loss of 4%. His total gain or loss percent is

10.A trader mixes 26 kg of rice at Rs. 20 per kg with 30 kg of rice of other variety at Rs. 36 per kg and sells the mixture at Rs. 30 per kg. His profit percent is

SIMPLE AND COMPOUND INTEREST

Principal: principal amount or sum is the money borrowed or lent out for a certain period.

Interest: Interest is the extra amount paid on principal amount with a year marked percent.Simple interest: If the interest is calculated uniformly on original principal amount through out the loan period, then it is called simple interest.Let principal = P Rate = R% per annum Time = T years.

Then, S.I. = PRT100 and Total amount A = P (1+ nr

100 )

Compound interest: Sometimes that the borrower and the lender agree to fix up a certain unit of time to settle the previous amount. In such cases, the amount after first unit of time becomes the principal for the second unit, the amount after second unit of time 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 the compound interest for the period.

1. When interest is compound annually, then the amount = P (1+ R100

¿¿n

2. When interest is compound Half – yearly, then the amount = P (1+(R/2)100

¿¿2 n

3. When interest is compound Quarterly, then the amount = P (1+(R/ 4)

100¿¿4 n

4. When interest is compound annually but time is in fraction, say n complete years and f fraction of

years, the amount = P (1+ R100

¿¿n (1+ fR100 )

5.When rates are different for different years, say R1%, R2%, R3% for 1st, 2nd, 3rd years respectively,

then the amount = P(1+R1

100 ) (1+

R2

100 ) (1+

R3

100 )

6. Present worth of Rs. X due n years hence is given by: present worth = x

(1+ R100

)n

7. Difference between C.I. and S.I. for 2 years = P( R100

)2

Problems :

1.A sum of money at simple interest amounts to Rs. 815 in 3 years and to Rs. 854 in 4 years. The sum is

2.  sum fetched a total simple interest of Rs. 4016.25 at the rate of 9 p.c.p.a. in 5 years. What is the sum?

3. How much time will it take for an amount of Rs. 450 to yield Rs. 81 as interest at 4.5% per annum of simple interest?

4. Reena took a loan of Rs. 1200 with simple interest for as many years as the rate of interest. If she paid Rs. 432 as interest at the end of the loan period, what was the rate of interest.

5. A sum of Rs. 12,500 amounts to Rs. 15,500 in 4 years at the rate of simple interest. What is the rate of interest

6. A lent Rs. 5000 to B for 2 years and Rs. 3000 to C for 4 years on simple interest at the same rate of interest and received Rs. 2200 in all from both of them as interest. The rate of interest per annum is

7. A man took loan from a bank at the rate of 12% p.a. simple interest. After 3 years he had to pay Rs. 5400 interest only for the period. The principal amount borrowed by him was:

8. The compound interest on Rs. 30,000 at 7% per annum is Rs. 4347. The period (in years) is

9. What will be the compound interest on a sum of Rs. 25,000 after 3 years at the rate of 12 p.c.p.a

10. At what rate of compound interest per annum will a sum of Rs. 1200 become Rs. 1348.32 in 2 years

11.If the simple interest on a sum of money for 2 years at 5% per annum is Rs. 50, what is the compound interest on the same at the same rate and for the same time

12. The compound interest on a certain sum for 2 years at 10% per annum is Rs. 525. The simple interest on the same sum for double the time at half the rate percent per annum is:

AVERAGES

Important Results:

1. Average = Sum of the different data / Number of the data.

2. When the same value ‘ a’ is added to each data or observation then new average is given by

New average = original average + a.

3. When the same value ‘ a’ is subtracted from each data or observation then new average is given by

New average = original average - a.

4. When the same value ‘ a’ is multiplied to each data or observation then new average is given by

New average = original average x a.

5. If ‘ x’ & ‘ y’ are the average of group ‘ a’ & group ‘ b’ resp. Then the combined average of two groups is

given by ax+bya+b .

PROBLEMS:

1.In the first 10 overs of a cricket game, the run rate was only 3.2. What should be the run rate in the remaining 40 overs to reach the target of 282 runs.

2. A family consists of two grandparents, two parents and three grandchildren. The average age of the grandparents is 67 years, that of the parents is 35 years and that of the grandchildren is 6 years. What is the average age of the family.

3. A grocer has a sale of Rs. 6435, Rs. 6927, Rs. 6855, Rs. 7230 and Rs. 6562 for 5 consecutive months. How much sale must he have in the sixth month so that he gets an average sale of Rs. 6500.

4. The average weight of 8 person's increases by 2.5 kg when a new person comes in place of one of them weighing 65 kg. What might be the weight of the new person

5. The captain of a cricket team of 11 members is 26 years old and the wicket keeper is 3 years older. If the ages of these two are excluded, the average age of the remaining players is one year less than the average age of the whole team. What is the average age of the team

6. The average monthly income of P and Q is Rs. 5050. The average monthly income of Q and R is Rs. 6250 and the average monthly income of P and R is Rs. 5200. The monthly income of P is

7. The average age of husband, wife and their child 3 years ago was 27 years and that of wife and the child 5 years ago was 20 years. The present age of the husband is.

8.The average weight of A, B and C is 45 kg. If the average weight of A and B be 40 kg and that of B and C be 43 kg, then the weight of B is.

9. A library has an average of 510 visitors on Sundays and 240 on other days. The average number of visitors per day in a month of 30 days beginning with a Sunday is

10. Find the Average of 4,10,16,22,28,34,40,46

ALLIGATIONS OR MIXTURES

1.In what ratio must a grocer mix two varieties of pulses costing Rs. 15 and Rs. 20 per kg respectively so as to get a mixture worth Rs. 16.50 kg.

2. . In what ratio water must be mixed with milk costing Rs. 12 per litre to obtain a mixture worth of Rs. 8 per litre

3. In what ratio must a grocer mix two varieties of tea worth Rs. 60 a kg and Rs. 65 a kg so that by selling the mixture at Rs. 68.20 a kg he may gain 10%

4. Find the ratio in which rice at Rs. 7.20 a kg be mixed with rice at Rs. 5.70 a kg to produce a mixture worth Rs. 6.30 a kg

5. The cost of Type 1 rice is Rs. 15 per kg and Type 2 rice is Rs. 20 per kg. If both Type 1 and Type 2 are mixed in the ratio of 2 : 3, then the price per kg of the mixed variety of rice is

6. A merchant has 1000 kg of sugar, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. The quantity sold at 18% profit is

7. A jar full of whisky contains 40% alcohol. A part of this whisky is replaced by another containing 19% alcohol and now the percentage of alcohol was found to be 26%. The quantity of whisky replaced is

Time and Distance

1.A person crosses a 600 m long street in 5 minutes. What is his speed in km per hour.

2. An aeroplane covers a certain distance at a speed of 240 kmph in 5 hours. To cover the same distance in 1 2/3 hours, it must travel at a speed.

3. If a person walks at 14 km/hr instead of 10 km/hr, he would have walked 20 km more. The actual distance travelled by him is

4. Excluding stoppages, the speed of a bus is 54 kmph and including stoppages, it is 45 kmph. For how many minutes does the bus stop per hour

5. A man complete a journey in 10 hours. He travels first half of the journey at the rate of 21 km/hr and second half at the rate of 24 km/hr. Find the total journey in km.

6. It takes eight hours for a 600 km journey, if 120 km is done by train and the rest by car. It takes 20 minutes more, if 200 km is done by train and the rest by car. The ratio of the speed of the train to that of the cars is.

7. A train running at the speed of 60 km/hr crosses a pole in 9 seconds. What is the length of the train.

8. A train 125 m long passes a man, running at 5 km/hr in the same direction in which the train is going, in 10 seconds. The speed of the train is

9. The length of the bridge, which a train 130 metres long and travelling at 45 km/hr can cross in 30 seconds, is.

10. A train passes a station platform in 36 seconds and a man standing on the platform in 20 seconds. If the speed of the train is 54 km/hr, what is the length of the platform.

11. A train 240 m long passes a pole in 24 seconds. How long will it take to pass a platform 650 m long.

12. Two trains of equal length are running on parallel lines in the same direction at 46 km/hr and 36 km/hr. The faster train passes the slower train in 36 seconds. The length of each train is.

TIME AND WORKImportant Results:

1. If A & B can do a pieceof work in ‘ a’ days and ‘ b’ days resp., then working together they will take ab / a + b days to finish the work and they will finish a +b / ab th part of the work in one day.

2. If more persons are employed, less days are required to complete a given job.3. If we want to complete a job in lesser number of days we will have to employ more persons.

(Assuming the same capability of a person and same number of hours of work each days).4. If more days are given to a group of persons, they will perform more amount of work.5. If a ipe fills a tank in ‘x’ hors and another pipe can empty it in ‘ y’ hours then the net part of the tank

filled in 1 hour = y –x / xy ( when both start working together) and the time required to fill the tank completely = xy / y - x hours.

6. A pipe can fill a tank in ‘x’ hours. But due to a leak at the bottomsince the beginning it takes ‘y’ hours. Then time taken by the leak to completely empty a filled tank of the same size = xy / y – x hours

PROBLEMS:

1. A can do a work in 10 days and B in 15 days. In how many days they together will do the same work?

2. A can do a piece of work in 4 hours; B and C together can do it in 3 hours, while A and C together can do it in 2 hours. How long will B alone take to do it.

3. A can do a certain work in the same time in which B and C together can do it. If A and B together could do it in 10 days and C alone in 50 days, then B alone could do it in.

4. A can finish a work in 18 days and B can do the same work in 15 days. B worked for 10 days and left the job. In how many days, A alone can finish the remaining work?

5. A can lay railway track between two given stations in 16 days and B can do the same job in 12 days. With help of C, they did the job in 4 days only. Then, C alone can do the job in.

6.A alone can do a piece of work in 6 days and B alone in 8 days. A and B undertook to do it for Rs. 3200. With the help of C, they completed the work in 3 days. How much is to be paid to C.

7.A can do a piece of work in 4 hours; B and C together can do it in 3 hours, while A and C together can do it in 2 hours. How long will B alone take to do it.

8.A can finish a work in 18 days and B can do the same work in 15 days. B worked for 10 days and left the job. In how many days, A alone can finish the remaining work.

9. A, B and C can complete a piece of work in 24, 6 and 12 days respectively. Working together, they will complete the same work in

10.

Twenty women can do a work in sixteen days. Sixteen men can complete the same work in fifteen days. What is the ratio between the capacity of a man and a woman

CLOCKS

The hour hand and the minute hand of a clock move in relation to each other continuously and at any

given point of time, they make an angle between 00 and 1800 with each other.

If the time shown by the clock is known, the angle between the hands can be calculated. Similarly, if

the angle between the hands is known, the time shown by the clock can be found out.

When we say angle between the hands, we normally refer to the acute/obtuse angles (upto 1800)

between the two hands and not the reflex angle (>1800).

For solving the problems on clocks, the following points will be helpful.

Minute hand covers 3600 in 1 hour, i.e., in 60 mins. Hence MINUTE HAND COVERS 60 PER

MINTUE.

Hour hand covers 3600in 12 hours. Hence HOUR HAND COVERS 300 PER HOUR.

The following additional points also should be remembered in a period of 12 hours, the hands make

an angle of

00with each other (i.e., they coincide with other) 11 times.

1800 with each other (i.e., they lie on the same straight line) 11 times.

900 or any other angle with each other 22 times.

Problems :

1.How many degrees does an hour hand move in 10 minutes.

2. How many degrees will the minute hand move in the same time, in which the hour-hand move.

3. At what angle are the hands of a clock inclined at 4 hours 20 minutes?

4. At what angle are the hands of a clock inclined at 20 minutes past 7?

5. At what time between 6 and 7 o’clock are the hands of a clock together?

6. Find the number of times the hands of clock are 180 degrees apart in a day?

7. Between two consecutive hours, how many times do both the hands of a clock be at right angles

8. Find the number of times the hands of clock are 360 degrees apart in a day

CALENDAR

Suppose you are asked to find the day of the week on 30th June, 1974, it would be a tough job to

find if you do not know the method. The questions based on calendars are sometimes asked in the

competitive exams. The method of finding the day of the week lies in the number of “odd days”.

NOTE: Every 7th day will be the same day count wise i.e. if today is Monday then Tuesday onwards

the 7th day will once again be Monday. Hence, logic says divide the total numbers of days by 7 and

the remainder will be called odd days.

Example: 52 days) 7=3 odd days.

Leap and Ordinary Year:

Non-leap year has 365 days whereas a leap year has one day extra be3cause of 29 days in the month

of February. Every year which is divisible by 4 is called a leap year. Leap year consists of 366 days,

(52 complete weeks + 2 days), the extra two days are the odd days. So, a leap year has two odd days

because 366) 7 =2 (remainder).

An ordinary year consists of 365 days (52 complete weeks +1 day), the extra one day is the odd day.

So, an ordinary year has one odd day.

Note: Every century, which is a multiple of 400, is a leap year.

Example: 400,800, 1200, 1600 .......are leap years.

Counting the number of Odd Days:

100 years consist of 24 leap years + 76 ordinary years. (100 years when divided by 4, we get 25 leap

years but 25th i.e. the 100th year is not a leap year, hence only 24 leap years)

=2x24 odd days + 1x76 odd days

=124 days

=17 weeks +5 days

The extra 5 days are the number of odd days.

So, 100 years contain 5 days.

Similarly, for 200 years we have an extra of 10 days (1 week +3 days).

200 years contain 3 odd days.

Similarly, 300 years contain 1 odd day and 400 years contain 0 odd days.

Problems :

1.What day of the week was 18th July 1978?

2. What day of the week would be 26th March 2023?

3. On which dates of March, 2008 will a Sunday come

4. If 4th August 1996 was a Sunday, then what day of the week was 23rd August 1959?

5. On which dates of march 2005 did Friday fall

Blood RelationsSynopsis:In this test, the success of a candidate depends upon his/her knowledge of blood relations, some of

which are summarized below to help solve these tests.

Mother’s or father’s son Brother

Mother’s or father’s daughter Sister

Mother’s or father’s brother Uncle

Mother’s or father’s sister Aunt

Mother’s or father’s father Grandfather

Mother’s or father’s mother Grandmother

Son’s wife Daughter-in-law

Daughter’s husband Son-in-law

Husband’s or wife’s sister Sister-in-law

Husband’s or wife’s brother Brother-in-law

Brother’s son Nephew

Brother’s daughter Niece

Uncle or aunt’s son or daughter Cousin

Sister’s husband Brother-in-law

Brother’s wife Sister-in-law

Grandson’s or Grand daughter’s daughter Great grand daughter

Remark : A relation on the mother’s side is called ‘maternal’ while that on the father’s side is called

‘paternal’. Thus, mother’s brother is ‘material uncle’ while father’s brother is ‘paternal uncle’.

1.Pointing to a photograph of a boy Suresh said, "He is the son of the only son of my mother." How is Suresh related to that boy?

2. If A is the brother of B; B is the sister of C; and C is the father of D, how D is related to A?

3. Introducing a boy, a girl said, "He is the son of the daughter of the father of my uncle." How is the boy related to the girl?

4. Pointing to a photograph says, "He is the son of the only son of my grandfather." How is the man in the photograph related to Lata?

5. My grandfather’s daughter-in-law’s husband’s son’s sister F is related to me as ?

6.E’s father’s only brother’s son’s wife’s brother-in-law is related to E as

7.S’s sibling’s grandfather’s only child’s grandson’s father’s father is related to S as

8. A goes to a picnic and meets a woman B who is the sister of A’s wife. How is B related to A.