How well do you know your arithmetic? I’m not just talking about adding, subtracting, multiplying and dividing, but a wide array of arithmetic operations, properties, and concepts that you’ll need to know for the Common Admissions Test (CAT). CAT arithmetic problems can be quite challenging, but if you study these practice exercises, then you’ll be better prepared for the real deal!

## CAT Arithmetic Topics

Here are some of the arithmetic topics that you may run across on the CAT:

- Mean, median, and mode (basic statistics)
- Percentage, ratios, and proportions
- Simple and compound interest
- Applications in profit & loss, speed, time & distance, and rates

### Basic Statistics

The study of statistics has to do with data and how it is distributed. Perhaps the simplest statistical measure is the **mode** of a data set.

#### Mode

The **mode** is the number that occurs most often in a data set. If no number occurs more than once, then there is no mode. On the other hand, a set can have more than one mode — that’s when more than one number has the maximum number of occurrances.

For example, the set (3, 1, 4, 1, 5, 9, 2, 6, 5) has two modes: 1 and 5.

#### Median

Next, the **median** is the “middle” number. In other words, if you arranged all of the data in increasing order, then the one in the middle of the list is your median. But this only works if there are an odd number of data points. For an even number, you have to take the two middle numbers, say *a* and *b*, and average them together to get the median: (*a* + *b*)/2.

#### Mean and Weighted Averages

And that brings us to the **mean**. The **arithmetic mean** of a set is equal to the sum of the data divided by the number of items. We also call this the **average** value.

So, the mean (average) of (*a*_{1}, *a*_{2}, *a*_{3}, …, *a _{n}*) is equal to: (

*a*

_{1}+

*a*

_{2}+

*a*

_{3}+ … +

*a*)/

_{n}*n*.

You’ll also need to know how to find **weighted averages**. This time, each value gets a *weight*, which could represent the number of times that value occurs, or the percentage of a population that corresponds to that value, or other useful interpretations.

Suppose the data set (*a*_{1}, *a*_{2}, *a*_{3}, …, *a _{n}*) has weights (

*w*

_{1},

*w*

_{2},

*w*

_{3}, …,

*w*). Then the (weighted) average is equal to:

_{n}For example, suppose your class has 15 boys and 18 girls. The test average for the boys was 78, while girls’ average was 81. What was the class average for this test?

The data points are 78 with weight 15, and 81 with weight 18. So we obtain: (15 × 78 + 18 × 81)/(15 + 18) = 2628/33 = 79.6.

### Percentage, ratios, and proportions

We use percents, ratios, and proportions to help understand fractional relationships among quantities.

#### Percentages

A **percentage** is measure of how frequent an item would be out of a hundred. So, *P*% really stands for the fraction *P*/100.

The *percentage change* between two values is equal to the *difference* divided by the *original*.

In order to find the new amount after applying a percentage *increase* or *decrease*, use the following formulas:

*P*% increase of*a*is: (1 +*P*/100) ×*a*.*P*% decrease of*a*is: (1 –*P*/100) ×*a*.

#### Ratios and Proportions

Ratios and proportions are other ways to express fractional relationships. The ratio *A* : *B* can be interpreted as a fraction *A*/*B*. If two ratios are supposed to be the same, then this sets up a proportion.

*A* : *B* = *C* : *D* means *A*/*B* = *C*/*D*. Equivalently, *AD* = *BC*.

### Simple and compound interest

Interest can be computed in a number of different ways. For the CAT arithmetic section, you’ll need to know simple and compound interest.

#### Simple Interest

**Simple interest** varies directly with the time of the investment. If my interest payment is $100 for a period of one month, then it would be $200 for a total of two months. Simple, right? The formula for computing simple interest is likewise very simple.

If *P* is the principle, *r* is the interest rate as a decimal, and *t* is the time in years, and *I* stands for the computed simple interest, then

*I* = *Prt*

Then to find the total amount of the investment, simply add:

*P* + *I* = *P* + *Prt* = *P*(1 + *rt*).

#### Compound Interest

Compounding interest means that the interest accrued in any given time period is then added back to the principle. That way, the interest computation for the next month becomes slightly larger. The process continues for the entire time of the investment. So compound interest generates much more in the long run than simple interest at the same rate.

Here is the formula for the total amount *A*, if there are *n* compounding periods per year. (Remember, your rate *r* must be the decimal equivalent of your percentage rate.)

### Applications of Arithmetic

There are numerous applications of arithmetic that show up on the CAT. Here are just a few.

- Profit = Revenue – Costs. If Profit < 0, then it is a loss. Revenue = (Price per unit) × (units sold).
- Distance = Speed × Time. Or, Speed = Distance / Time.
- Rates are measures of change in one quantity per change in another. Therefore, a rate is like a ratio, or fraction. Rate of A with respect to B = (Change in Quantity A) / (Change in Quantity B).

## CAT Arithmetic Practice

Now let’s see if you can apply what you know about arithmetic on the following problems! Solutions will be given at the end.

### Problem 1

The number of part-time employees at a large chain of retail stores changes from month to month. Use the following chart to determine by what percent the part-time workforce increased from March to April.

Month | Jan. | Feb. | March | April | May |

Part-time Employees | 320 | 308 | 358 | 402 | 357 |

(A) 9.3% (B) 10.1 % (C) 12.3% (D) 15.7%

### Problem 2

A company has two factories, A and B, both producing the same model of smart phones. The ratio of production between factories A and B is 2 : 5, and the ratio of costs generated between the two factories is 3 : 4. What is the ratio of cost per unit between factories A and B?

(A) 2: 7 (B) 18 : 13 (C) 6 : 11 (D) 15 : 8

### Problem 3

There were three assignments in Writing Class, a 5-page paper worth 20% of the total course grade, a 10-page paper worth 30% of the course grade, and a final 20-page paper worth 50% of the course grade. If your scores were 82% for the 5-page paper, 65% for the 10-page paper, and 91% for the 20-page paper, what was your final course score to the nearest percentage?

(A) 81% (B) 79% (C) 77% (D) 75%

### Problem 4

Three brothers, Alan, Bobby, and Charles, work for a house-painting business. If working alone, Alan can complete one house job in 15 days, Bobby can do it in 20 days, and Charles takes 30 days to complete it. For a particular job, they work together for some number of days, after which Charles gets sick and cannot work, leaving Alan and Bobby to complete the project. The total amount of money received by the crew was $4,500, and Charles got $1500 less than either of his brothers. Assuming the daily rate of pay was the same per person, find out how many days the job took in total.

(A) 4 (B) 6 (C) 8 (D) 10

### Solution to Problem 1

(C) 12.3%

This is a job for the *percentage change* formula. Take the difference of the March and April numbers: 402 – 358 = 44. Then divide by the original (March) value: 44/358 = 0.1229. Finally, convert to a percentage to answer the problem. Roughly 12.3%, which is letter (C).

### Solution to Problem 2

(D) 15 : 8

This problem is extra challenging due to the apparent lack of information. The problem does not give any concrete numbers for total production or cost, only their ratios. However, there is just enough information to answer the question!

First, for production, we can say that the total production for factory A is 2*x* and for factory B is 5*x*. What is *x*? Surprisingly, it doesn’t matter… just think of “*x*” as an unspecified number of “parts.”

We do the same for costs, only we should use a different letter for the number of “parts.” Costs for factory A: 3*y*, and for factory B: 4*y*.

Now cost per unit is equal to the total cost divided by the total production. For A, we get: (3*y*)/(2*x*). For B, we find: (4*y*)/(5*x*). The problem asks for the ratio of these two quantities, so let’s treat the ratio as a fraction.

The answer is (D) 15 : 8.

### Solution to Problem 3

(A) 81%

This problem requires a weighted average. Here, the weights add up to 1 (20% + 30% + 50% = 100%), so we just have to compute the top part of the weighted average formula:

(82)(0.20) + (65)(0.30) + (91)(0.50) = 16.4 + 19.5 + 45.5 = 81.4.

Therefore, to the nearest percent, the course score is 81%.

### Solution to Problem 4

(C) 8

By far, this is the most complicated question on this practice set! There are numerous quantities to keep track of, and it isn’t clear how they all might be related.

First, think of the individual job completion times in terms of rates. The rate equals 1 job / *x* days. So Alan’s rate is 1/15, Bobby’s rate is 1/20, and Charles’ rate is 1/30.

Next, we must distinguish the work done when all three brothers were working (say, a fraction *p* of the total house job) from the when only two worked (1 – *p* of the total job).

Now let’s talk about the payments. Because Charles lost out on $1500 compared to each of his brothers, that means that the two other brothers earned $3000 for those days when Charles was missing, which leaves $1500 in payments when all the brothers were working, or $500 per brother. Given the ratio of $500 : $1500 for per-worker pay, we know that the two remaining brothers were on the job for three times the number of days as when all three were working.

Let *T* be the number of days when all three worked. Then Charles was absent for 3*T* days beyond that time.

Lastly, we must put everything together. Using Quanity = Rate × Time, we may observe:

- Before Charles left:
*p*= (1/15 + 1/20 + 1/30) ×*T*= (3/20)*T* - After Charles left: 1 –
*p*= (1/15 + 1/20) × 3*T*= (7/20)*T*

Then using Equation (1) in Equation (2), we find:

1 – (3/20)*T* = (7/20)*T*

1 = (10/20)*T*

*T* = 20/10 = 2

Therefore, the total time on the job was *T* + 3*T* = 2 + 6 = 8 days.

## More Information

For more practice, try this: Profit and Loss Practice Problems: CAT Quant Sample Question and Answer.

If you’d like to learn more about what is covered on the CAT, check out What is the CAT Syllabus?

Happy studying!

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