# Arithmetic for CAT – A Detailed Guide with Examples (Updated)

## Arithmetic is the branch of mathematics concerned with the study of numbers, their properties, and results of the application of operations on numbers, that is, addition, subtraction, division, and multiplication. Arithmetic is one of the most diverse subject matters in the QA section, consisting of topics that are of interest for everyday use, in addition to utility in business.

Arithmetic topics range from basic properties of numbers; to advanced operations and formulas; and further consist of topics such as percentages, profit and loss, interest, averages, mixtures, and so forth. Following are some of the important concepts for CAT under arithmetic:

**Percentages**– The word ‘percent’ can be understood as part per 100. Percentage expresses a number as a fraction of 100. It is a dimensionless number and is a representation of a part of the whole. The percentage is calculated by dividing the given ‘part’ or portion (numerator) by the whole (denominator) and multiplying by 100. The percentage is a very important and useful concept for CAT, both for QA and LRDI section, and therefore, the ability to compute quick percentages; to convert percentages to decimals, and fractions to percentages come in handy. The various concepts of percentages are as follows

**Percentage of**– As specified above, it presents a given quantity as a fraction of the total. For example, in a class of 100 students, if 45 are girls, the percentage of girl students of the total is 45/100*100 = 45%.**Percentage more or less**– It expresses by what percent one quantity is greater than or less than another. For example, if the pocket money of A is 500 and B is 400, A’s is 100/400*100 = 25% more than B, and B’s is 100/500*100 = 20% less than A’s.**Percentage increase or decrease**– It expresses by what percent a quantity has increased or decreased during a period under consideration, or compared to another quantity. For example, if the price of a top is now 500 when earlier, it was 400, it has now increased by 100/400*100 = 25%.**Cumulative Change**– It is the accruing change in a quantity, expressed as a percentage. For example, a car purchased for 25,000, depreciates by 10% in value each year, its value after two years would be 90%*90%*25000 = 20,250.

As can be seen, percentages are applicable in all spheres; prices, income, expenses, production, and so on, and are useful in the LRDI section as well where the questions frequently require the applicants to either analyze, compute or compare percentages, and thus, the applicants should be familiar with the concepts and tricks for the same.

**Profit and Loss**– This concept is a part of middle school mathematics and is useful, for all managers and non-managers alike. The basic concept states that profit refers to the gain to the seller when he sells an item at a price that is more than he purchased it for, and loss is the loss incurred when he sells the item at a price less than what he purchased it for. The various concepts Selling Price, Cost Price, Profit, and Loss, are familiar to all. Other concepts that the applicants should brush up on are Marked Price, the price mentioned on the item and the discount, or the reduction in the Marked Price, which gives the actual Selling Price, gain, or loss in case of wrong scale/measurement, freebies, and equivalent discount. The topic of profit and loss is vast, and consists of a lot of concepts, but is relatively easier to understand and thus, applicable.

**For Example**, A sells 16 sheep at a gain of 12.5% and 20 more at a certain gain percent, if he gains 25% overall, to determine the latter gain percent, Assume the CP of all 36 be 100 each which gives CP as 3600. 16 were sold at a gain of 12.5% or 112.5% of the CP, that is, SP of 16 is 1800. Assuming the gain on the remaining 20 as x%, SP is 100+x% 0f 2000. Total SP is given by 1800 + 20 (100 + x), which is 25% more than 3600 or 125% of 3600.

1800 + 2000 + 20x = 4500, 20x = 4500 – 3800; 20x = 700; x = 35

**Therefore, the latter gain percent is 25.**

**Simple and Compound Interest**– Interest is the money charged by or received by a lender for lending a certain sum for a given period. The basic concept behind interest is that the value of a unit of money does not remain constant. It decreases over a period due to inflation, which refers to the fall in the purchasing power of money. In other words, an amount has more value today, than in the future, and thus, a lender charges interest for the fall in the worth of the money lent. Other factors for charging interest include the risk of default by the borrower and thus, the rate of interest depends upon the perceived risk. Also, the lender lending to one borrower forgoes the opportunity to make other investments, and thus, the interest is also determined by the opportunity cost. Simple interest is the interest that remains constant over each year during the period or, the interest accrues only on the principal. Compound interest refers to the interest which accrues, on the principal as well as the interest of the previous year. The concepts are important, not only for money lent and borrowed but for cases such as changes in population or experiments and so forth.

For Example, A earns an interest of 600 over two years at simple interest and 630 over compound interest. To determine the rate of interest,

In the case of SI, the interest remains the same, interest for one year = 600/2 = 300,

For CI, the first-year interest is equal to SI, from the second year, the interest is given on the initial principal and the interest from the previous year, or the difference of 30 is from the second year.

Interest is thus, 30 on 300 for one year, rate = 30*100/300 = 10%

**Averages**: Being the most common topic since middle and high school maths and, with obvious practicability, the concept of averages is one such concept with which all applicants are familiar. Simply put, the average is the sum of observations divided by the number of observations. It comes in handy in all sorts of situations, determining per person the price of a ticket, per kg value of a commodity, average increase or decrease in sales, and so forth. Also, in the test, questions on averages appear both for QA and DI, be it an interpretation of a graph, chart, or comparison of a given set of periods or otherwise. Being aware of the various formulas, shortcuts and concepts would thus be agreeable for the applicants. Weighted average refers to the average for a given set of terms when each term either has a different number of occurrences, or varying values for a situation or disparate features, age, height, and so on. The concept of averages also comprises measures of dispersion, a value that represents the extent of scatter or dispersion of the given variables from the average.

**For Example**, The average weight of the lightest 4 students in a class of 5 students is 40kg and that of the 4 heaviest is 45kg, to evaluate the difference in the maximum and minimum possible average,

For the minimum total average; It is given that the average of 4 lightest students is 40 or it can be assumed each of the four weighs 40kg and the heavier one weighs 45kg; Class Average = 40*4+45/5 = 41

For the maximum possible average, it is given that average of 4 heaviest students is 45kg or each of the four weigh 45kg and the lighter one weighs 40kg; Class Average = 45*4+40/5 = 44

**Difference = 44-41 = 3**

**Mixtures and Alligations**: This concept can be applied in the majority of topics for CAT, not only for QA, but for DI as well, and is fairly simple to grasp, and should not be overlooked. Mixtures are obtained by the mixing of two or more components in a specific ratio. The concept of allegation is used to find the average of values that are composed of varying elements or elements in diverse ratios, as against weighted averages, where the elements are multiplied by factors of importance. It can also be used to find the ratio of quantities of the values when the average is known. As mentioned, alligations can be applied to find the average speed, time, price, profit or loss, quantity in case of compound mixtures, etc. This application of allegation gives fairly quick results and can thus, save time during the exam. The candidates should cover this topic thoroughly, as the concept is useful not only for the test but for everyday issues as well.

**For Example**: To determine the proportion in which tea at 7.5 per kg should be mixed with tea at 10.5 per kg to produce a mixture worth 8.5 per kg,

Mean price = 8.5; Cheap price = 7.5 and dear price = 10.5

Quantity Ratio = 10.5 – 8.5/8.5-7.5 = 2/1 or 2:1

**Ratio, Proportion, and Variation**: Ratio refers to the fraction one value is in terms of another, or the number of times one value is in comparison to another. The ratio is represented in the form a:b where a or the numerator is antecedent and b or denominator is consequent. Ratios are used to compare two quantities of the same kind, that is, the units of measurement should be the same. The ratio itself is an absolute number, without any unit of measurement, as it is a measure of comparison of quantities. Ratio being a fraction, essentially remains the same (simplest form), even if antecedent and consequent are multiplied or divided by the same non zero number, which means, the ratio in which two quantities are mixed remains the same even on change in the quantity of the mixture or compound. When the ratio of two terms is equal to the ratio of two other terms, the terms are said to be in proportion. These are represented by a:b:: c:d, where a, b, c, and d are referred to as first, second, third, and fourth proportional respectively. The terms a and d are called extremes while b and c are called means and the product of extremes is equal to the product of means, ad = bc. When two quantities, a and b, are such that an increase or decrease in one is accompanied by the same effect in the other and vice versa, the two quantities are said to be in direct proportion or variation. On the contrary, if the increase or decrease in one quantity is accompanied by the reverse effect in the other, and vice versa, the two quantities are said to be in inverse proportion or variation.

**For Example**, The ratio of ages of a couple is 4:3, four years hence, the ratio would be 9:7. At the time of their marriage, the ratio was 5:3, the number of years since the marriage can be determined as

The present ages are in the ratio 4:3, assuming the ages as 4y and 3y, in four years, ratio would be 9:7 or 4y+4/3y+4 = 9/7; 28y+28 = 27y+36; y = 8

Present ages, therefore, are 32 and 24 years respectively

Given, the ratio at the time of marriage was 5:3; assuming several years since the marriage as x,

32-x/24-x = 5/3; 96-3x = 120-5x; x= 12

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**Thus, the couple got married 12 years ago.**

**Partnership**: A partnership is an association of persons who contribute money to run a business and, share in the proceeds from that business. The partners managing the business are called working partners and the ones merely contributing capital and sharing proceeds are called sleeping or dormant partners. The concept of partnership is based on ratio and proportion. The share of partners is determined by dividing the earnings of the business in the ratio of contribution of the partners, or case of varying periods for each partner, in that of the equivalent investment of the partners, or the time contributed by the partners in managing the business. The questions on a partnership may be about the distribution of the profit/loss among the partners, or computation of total earnings when the share of partners are given, or a comprehensive set with varying periods, partners, and their investments.

**Time and Work**: This concept relates to almost all spheres of work efficiency, and as future managers, the applicants should understand the correlation between the work to be done in the given period and with accuracy. This concept is based on the premise that the ‘resources’ employed, that is time in terms of hours worked or several days to finish, several people, in doing the ‘work’, which may be the construction of a house, completing an assignment, etc, and the ‘output’ that is, the result are directly proportional. The more of the resources are employed, the faster or more of the output can be obtained. Another concept here is that of efficiency, that is the amount of work a person can do per unit of time. For Example, if A can finish a task in 10 days, A would do 1/10^{th}of the work in one day. And B can do the same task in 12 days, so B would do 1/12^{th}of the task in one day. The questions might of the form to determine the time in which A and B would finish the task working together, separately, alternately, or when either of them fails to continue. The questions may also be based on the number of people required to finish a task within a given period or the amount of work done by a given number of workers in the given period or comparing the efficiency of two or more workers. The concept finds at least ten percent questions in the CAT quant and should, therefore, be focused on.

**Pipes and Cisterns**: The concept of pipes and cisterns is an extension of that of time and work. Here, the work done can be taken as the pipes being used to fill or drain, and their efficiency in doing so. For example, a 3000-liter tank has three pipes, A, which can fill the tank in 12 hours, B which can empty it in 14 hours, and C which can fill it in 11 hours. The questions might be based on the time taken to fill or empty an already filled tank, or where only the pipes are given the capacity of the tank, and so forth. The rate or efficiency of the pipes should be taken as positive or negative based on whether the question requires the tank full or empty, that is, in case of full, the efficiency of the pipes which fill up the tank is taken positive and that of the drainpipe is taken as negative, and vice versa in case of emptying a tank. For Example, a tank has a pipe that can fill it in 10 hours, due to a leak in the tank it takes five hours more to fill the tank, in how many hours can the leak empty the tank? Assuming the capacity of the tank is 100 liters, the pipe, say A can fill it in 10 hours or it can fill 100/10 = 10 liters per hour. Assume the efficiency of the leak as x liters per hour. According the statement, 100/10-x = 15, or 100 = 150 – 15x, 15x = 50 or x = 50/15, time taken by the leak to empty the tank would be thus, capacity of the tank divided by the rate of B, 100*15/50 = 30 hours.

**Time, Speed, and Distance**: The questions from the concept of time, speed, and distance are a norm for entrance tests, at least a couple of questions can be seen each year. The basics of the concept of time, speed, and distance are part of the curriculum for both middle school maths and physics. The concepts here a tad more detailed with average speed, the relation between speed, time or distance, speed or time, distance; when the third one is constant, the concept of linear and circular races, and so forth. Speed refers to the rate at which one covers a particular distance. From the formula, speed = distance/time, it can be seen that when time is constant or the same for any given situation, speed and distance are directly related, or the distance traveled is in the same ratio as the speed. Similarly, if speed is constant, distance and time are directly related. But, when distance remains the same, the speed and time are inversely proportional or higher the speed, lower is the time taken to cover the given distance and vice versa. The concept of average speed refers to the total distance covered in the total time taken, (irrespective of the different modes of traversing). Relative speed refers to the speed of one about that of another, it depends on the direction of travel. For example, in a situation where you are traveling in a car, there is a car from the opposite direction, when the other car crosses yours, the relative speed is equal to the sum of the two speeds. If it is moving in the same direction, the relative speed is equal to the difference in the two speeds. These concepts are the basis for the concept of races, which are the gist of a lot of questions. Moreover, the concept also entails upstream and downstream, the speed of a boat about that of the water, the concept of elevators, and so forth. Though the concept of time, speed, and distance is a vast one, questions on this topic are present in almost all entrance tests. Attempting even a couple of questions accurately from time, speed, and distance thus goes a long way.

For Example, A travels half of his journey by bus at the speed of 200/9 m/s and half by metro at 120 km/h, for calculating the average speed for the entire journey, the speeds should be in the same unit

200/9*18/5 = 80 km/hr,

Since the distance was the same (half), the average speed is given by the harmonic mean of the two speeds. Average speed = 2*80*120/80+120 = 96 km/hr

- Clocks and Calendars: The concept of clocks and calendars can be said to be related to that of time, speed, and distance, with a lot of questions of clocks focused on finding the time at which the minute and hour hand or minute and seconds hand are at a particular angle or coincide and so on. The other questions in this topic may relate to faulty clocks and thus, determining the correct time, and in case of calendar-related questions, determining the day for a particular date based on the day given in the previous year, several leap years in a given period, number of Mondays in a given period, etc. The concept of the angle between the minute and hour hand or minute and the second hand is based on relative speed, that is, the time taken by the minute hand to cover the given angle relative to the hour hand or second hand relative to the minute hand. The minute and hour hand coincide 22 times in twenty-four hours, as it concurs between 11 and 1’o clock at 12, after every 720/11 minute.

Above is a summary of some of the important concepts of Arithmetic, for entrance tests for the various management courses and institutes, and the aspirants for the same should start from basics, attempt practice, and previous year questions and mock papers.