Quantitative
A full review of quantitative ability for entry tests: test-taking strategy, basic arithmetic, number sets, fractions, decimals, algebra, expressions, equations and simultaneous equations, with worked examples and practice.
The best way to study Mathematics for the test is to make a list of rules and principles given in the review of quantitative section and explanation of the answers to the practice tests found later in this book. Keep a sheet of paper handy, as you are reviewing these tests; jot down any principles, rules, definitions, or formulas that are unfamiliar to you. Then take time to memorize those rules. It is critical that you be able to recall automatically a formula, a short-cut tip or suggestion so you do not spend valuable time on the exam working out a lengthy problem or "re-inventing the wheel" each time. Use the review section for added reinforcement.
FIRST: Read the entire problem. Do not start doing calculations until you have read the problem from start to finish; you may find that you do not need to do all the work. For example, if the problem states to find the value of the expression (n - 13) (n - 52)(n - 33)(n - 4), for n = 4. You know automatically that the answer is 0. This is because in the fourth parentheses the value of n - 4 for n = 4 is 0 and anything multiplied by 0 is 0. You saved all the calculations involved.
SECOND: Read the answer choices. Often you can narrow your answer down by estimating or by looking carefully at the units or values. For example, you may be asked the number of soldiers in a platoon as a captain asks to stand all his soldiers in the form of a square. Looking at your answer choices, you see 108, 196, 224, and 87. Only the second answer choice could be correct, since it is a complete square.
THIRD: Make absolutely certain that you know exactly what the question is calling for. Many careless errors are made when a person solves for x, not noting that the question asks for y, or for 2x. You should take at least as much time reading and thinking about the problem as you actually do on calculations.
Example: A car goes from city M to city K, 200 miles apart, at an average speed of 50 miles per hour. How many miles the car will be from city K after 3 hours. Answer Choices: (A) 100 miles (B) 20 miles (C) 50 miles (D) 150 miles. The car travels 150 miles in 3 hours (50 x3). So it will be 150 miles from city M, but the question asks the distance from city K. The right choice is C.
FOURTH: If none of the known methods works, try grabbing an answer from the four options and plugging it into the question. This will often lead you to the right answer quickly.
Example: During the first month after the opening of a new shopping home, sales were $72 million. Each subsequent month, sales declined by the same fraction. If the sales during the third month after the opening totaled $18 million, what is that fraction? Answer Choices: A 1/4 B 1/4 C 1/4 D 2/3. The fastest way to a solution is to plug in an answer. Try choice A. If sales in the second month after opening are declined by 1/4, it gives the sales of $18 million but this figure is for third month after the opening, so this is not the right choice. Drop it. Only the choice C gives the right sale figure during the third month. Therefore, the right choice is C.
Many math questions in NAT are solved by using signed numbers, algebraic manipulation, equations and inequalities. Important rules and concepts will now be reviewed.
Some questions involve arithmetic calculations and require algebraic simplification. Recall some basics from your memory.
7 - 5 = 2, 5 - 7 = -2 5x - 7x = -2x, 7x - 5x = 2x 3x^2 - 5x^2 = -2x^2, 6x^3 - 2x^2 = 6x^3 - 2x^2 cannot be added as the exponents are different
All positive and negative whole numbers including zero are integers.
Example: -349 , -1 , 0 , 4 , 77 , 183
Any number that cannot be divided by 2.
Example: 3, 5, 7, 11, 33, 45 etc
Any number that can be divided by 2.
Example: 2, 4, 6, 80, 96, 110 etc.
The number that can be divided by itself and by 1 only; there is no other factor of the number.
Example: 2, 3, 5, 7, 11, 13, 17, 19 etc. 2 is the only even Prime number. 1 is not a prime number.
Product of two prime numbers can never be a prime number. 3 × 7 = 21, which is not a prime number as it is divisible by 3 and 7. Sum (Addition) of two prime numbers may or may not be a prime number. 2 + 3 = 5, 2, 3 and 5 all are prime numbers. 2 and 7 are prime numbers but their sum 9 is not a prime number. 2 + 7 = 9, 2 and 7 are prime numbers.
The absolute value of a quantity is the quantity with only positive value. The symbol for absolute value is two enclosing vertical segments. The absolute value of -11 and +11 is written as |-11| and |+11| and is equal to 11.
If you add two quantities that have the same signs, simply add them and retain the sign. 2 + 7 = 9 and -2 + (-7) = -9. If you add two quantities of different signs, simply add them and the result will bear the sign of the quantity of greater absolute value.
2 - 7 = 2 + (-7) = -5 7 - 2 = 7 + (-2) = 5 3A + 4A = 7A 3A - 4A = -A 7a^2 + 2a^2 = 9a^2
3a^2 - 2a can't be added since they are not like terms. 3x + 5 can't be added since they are not like terms. If three or more quantities are to be added, add like positives, then like negatives, then combine like terms by subtracting absolute values.
8 + 7 -13 +12 = +27 -13 = +14 6c - 5d -4 -8c +7d - 6 = -2c + 2d -10
If two quantities having the same sign are multiplied, the answer is positive (+). If two quantities having the different signs are multiplied, the answer is negative (-). Two parentheses with no sign between them indicates multiplication, No sign between a quantity and a parenthesis also indicates multiplication. A raised dot between two quantities indicates multiplication as well.
(+6)(+5) = 30 (-9)(-3) = +27 -7(8) = -56 4 · -4 = -16
You can only add or subtract like terms. However, all terms, whether alike or different, can be multiplied. When like letters are multiplied, add exponents.
3(-2Y) = -6Y (7G)(-3K) = -21GK A^3 × A^4 = A^7
Dividing two quantities having the same sign, the answer will be positive. Dividing two quantities having different signs will give negative answer.
-18/-3 = +6 -18/+3 = -6 +18/+3 = +6 +18/-3 = -6
Dividing like letters, subtract their exponents. If a letter such as Y bears no exponent it has exponent of 1.
Y^4/Y^3 = Y Y/Y = 1 8Y^8/2Y^2 = 4Y^6
When a quantity divides another quantity that contains two or more terms, divide each of term by the first quantity.
(6P - 10)/(-2) = -3P + 5 (12Y + 6)/6 = 2Y + 1 (5x - 7)/3 = (5/3)x + 7/3 (5b^2 + 10b)/5b = b + 2
sqrt(4) = sqrt(2×2) = 2 , sqrt(x^2) = sqrt(x × x) = x
Many problems involve multiple operations, The operations must be performed in a particular order. Occasionally test makers like to see whether you know what that order is. Here's an easy way to remember the order of operations: Please Excuse My Dear Aunt Salma.
This stands for:
| Letter | Operation | |
|---|---|---|
| P | for | Parentheses |
| E | for | Exponents |
| M | for | Multiplication |
| D | for | Division |
| A | for | Addition |
| S | for | Subtraction |
Do operations enclosed in parentheses first: then take care of exponents: then you multiply, divide, add, and subtract in the sequence, moving from left to right.
A fraction is just another way of expressing division. The expression 12/17 is exactly the same thing as 12 divided by 17. a/b is nothing more than "a" divided by "b". In the fraction x/y, x is known as numerator, and y is known as the denominator. The other important way to think of a fraction is as Part/Whole. The fraction 7/10 can be thought as 7 parts out of a total of ten parts of an item (number value).
PROPER FRACTIONS: If the Numerator is less than Denominator in a fraction, the fraction is called a proper fraction.
IMPROPER FRACTIONS: If the Numerator is greater than Denominator in a fraction, the fraction is called an improper fraction.
MIXED FRACTIONS: Mixed fraction is a combination of a whole number and a fraction. Like in 7*(2/5), 7 is wholenumber and 2/5 is the fraction.
To add two or more fractions that have the same denominator, simply add up the numerators and put the sum over the common denominator.
Example: 1/7 + 5/7 = (1+5)/7 = 6/7
Subtraction works exactly the same way: 6/7 - 2/7 = (6-2)/7 = 4/7
Before you add or subtract two or more fractions with different denominators, you must give all of them the same denominator. To do this, multiply each fraction by a number that will give it a denominator in common with the others.
Example: If you wanted to change 2/1 into sixths, you could do the following: 1/2 × 3/3 = 3/6. We haven't actually changed the value of the fraction, because we multiplied it by 1. The new fraction reduces to 1/2.
If we wanted to add 1/2 + 2/3 = (1/2 × 3/3) + (2/3 × 2/2) = 3/6 + 4/6 = 7/6
To multiply fractions, just and put the product of the numerators over the product of the denominators.
Example: 2/3 × 4/5 = 8/15
When you add or multiply fractions, you often end up with a big fraction that is hard to work with. You can usually reduce such a fraction. To reduce a fraction, find a factor of the numerator that is also a factor of the denominator, it saves time to find the biggest factor they have in common, but this isn't critical. You may just have to repeat the process a few times. When you find a common factor, cancel it. For example, 2/3 ÷ 3/4. Get used to reducing all fractions (if they can be reduced) before you do any work with them. It saves a lot of time and prevents errors in calculation.
To divide one fraction by another, just invert the second (from left to right) fraction and multiply: 2/3 ÷ 3/4 is the same thing as 7/2.
The Test sometimes gives you numbers that are mixtures of integers and fractions, for example, 3*(1/2). It's easier to work with these numbers if you convert them into mixed fractions. 3*(1/2) would be converted like this: Multiply 2 with 3 and add 1, this gives you the numerator of the resulting mixed fraction. The resulting mixed fraction is 7/2.
In the course of a problem you may have to compare two or more fractions and determine which is larger.
Example: Compare 3/4, 7/8, 9/11, which one is the greatest fraction. First take two fractions 3/4, 7/8 multiply numerator of the first with the denominator of the other (3×8 = 24), similarly multiply numerator of the second with the denominator of the first (7 × 4 = 28). Since 28 is greater than 24 so 7/8 is greater than 3/4. Now take 7/8, 9/11 to compare. Multiply numerator of the first with denominator of the second (7 × 11 = 77), similarly multiply numerator of the second to the denominator of the first (9 × 8 = 72). Since 77 is greater than 72, so 7/8 is the greatest of the three given fractions. To compare 8/15 and 7/9, multiply 8 with 9 and 15 with 7, Since the product of 15 and 7 (105) is greater than the product of 8 and 9 (72), so 7/9 is greater than 8/15.
Tips: If the difference between numerators and denominator of one fraction is the same as the difference between the numerator and denominator of the other fraction, then the fraction with greater numerator is greater. In case of 3/4, 7/8, the difference between 3 and 4 is same as difference between 7 and 8. Since 7 is greater than 3, so 7/8 is greater fraction.
When decimals are added or subtracted, the decimal points must be placed one under the other. Every integer has its decimal after unit digit (45 is the same as 45. and $45 is the same as $45.00).
Example: 4.9 + 0.73 + 7.00 Line up the decimal points. To add them, fill the empty spaces zeroes, add as usual and place the decimal point in the line of the decimal points of the numbers to be added. 4.9 + 0.73 + 7.00 = 12.63
Example: Which is the largest, 0.073, 0.5, 0.586, 0.08, or 0.59? Place the numbers under one another, lining up the decimal points. Fill in zeroes so that all of the decimals have the same number of decimal places. 0.073, 0.500, 0.586, 0.080, 0.590. 0.590 is the largest three-place decimal. Answer: 0.59.
When you multiply decimal numbers, the decimals do not have to be under one another. The product (answer) must contain as many numbers after its decimal as the total of the decimal places in the two numbers being multiplied. For example, find the product of 0.28 and 0.3. 28 times 3 is 84, but where should the point be placed? 0.28 has two numbers after its point and 0.3 has one number after its point, making a total of three decimal places. Count three places to the left from the end of 84. Since 84 have two places, a zero must be placed in front of 8. Answer: 0.084
1. Arrange in descending order: 3/8, 4/9, 2/7. Make the comparisons of two fractions each. 4/9 is greater than 3/8, and 4/9 is greater than 2/7, so 4/9 is the largest fraction. 3/8 is greater than 2/7, so the answer is 4/9, 3/8, 2/7.
2. 40 + 80/0.4 = what number? First simplify the fraction: 80/0.4 = 80/0.4 × 10/10 = 800/4 = 200. Then add: 40 + 200 = 240.
3. If a bushel of apples weighs from 48 to 54 pounds and a bushel of melons weighs from 80 to 90 pounds, what is the smallest ratio between the weight of a bushel of apples and a bushel of melons? "Smallest ratio" means smallest fraction, which will contain the smallest numerator but the largest denominator. The answer is 48/90, or 8/15.
4. Simplify: (3*(1/3)) / (5*(1/3) + 6*(1/3)). Add the fractions in the denominator: 10/3 ÷ 35/3 = 10/3 × 3/35 = 2/7.
5. If r is greater than 0 and b = 1/r, does b increase or decrease as r increases? If the numerators are the same, the smaller fraction has the larger denominator. Therefore, as r increases and the numerator remains 1, the fractions get smaller and b decreases.
6. Reduce 12c^2 / 15c. Reduce 12 and 15 by canceling both by 3. Treat c^2 / c = c^(2-1). The answer is (4/5) c.
7. Add m/2 + m/3. Find L.C.D., which is 6. Convert each fraction to sixths and add: (m/2 × 3/3) + (m/3 × 2/2) = 3m/6 + 2m/6 = 5m/6.
8. Subtract 2/5x from 3/4x. In a subtraction example, the quantity after the word "from" goes first: 3/4x - 2/5x. L.C.D. is 20x. (15x - 8x)/20x^2, 15x - 8x = 7x. So the answer is 7/20x.
9. A woman owned 2/3 of a store and sold 1/5 of her share. What part of the store did she still own? "1/5 of her share" means 1/5 times her share. 1/5 of 2/3 = 2/15 was sold. 2/3 - 2/15 = (10-2)/15 = 8/15 is the answer.
10. Change 0.68 to a fraction. 0.68 = 68/100 = 17/25.
11. Change 3/16 to a decimal correct to the nearest thousandth. Carry the division one place past thousandths: 3/16 = 0.1875. Since the extra place is a 5, we round up. The answer is .188.
12. (1/6)(5/6) = x times (4/9). 5/36 = x × 4/9. Divide by 4/9: 5/36 ÷ 4/9 = 5/36 × 9/4 = 5/16.
13. If 5x = 28, what does 3x equal? x = 28/5. Multiply both sides by 3: 3x = 84/5 or 16*(4/5).
14. If 4y + 12 = -30, what does (y + 2) equal? 4y = -42, y = -21/2. y+2 = -21/2 + 4/2 = -17/2.
15. If r = 3b, what does (3/4) r equal? Multiply both sides by 3/4: (3/4) r = (3/4)(3b) = 9b/4.
| Q | Ans | Q | Ans | Q | Ans | Q | Ans | Q | Ans |
|---|---|---|---|---|---|---|---|---|---|
| 1 | C | 2 | A | 3 | B | 4 | C | 5 | D |
| 6 | D | 7 | A | 8 | C | 9 | C | 10 | D |
| 11 | A | 12 | D | 13 | D | 14 | D | 15 | D |
| 16 | A | 17 | D | 18 | B | 19 | D | 20 | A |
| 21 | B | 22 | B | 23 | B | 24 | A | 25 | B |
| 26 | A | 27 | D | 28 | D | 29 | D | 30 | D |
| 31 | A | 32 | D | 33 | A | 34 | A | 35 | A |
| 36 | D | 37 | A | 38 | C | 39 | A | 40 | B |
Algebra is the art of dealing with unknown numbers.
A variable is an unknown number, also known simply as an unknown. You represent a variable with a single letter, such as x or y. When you see x, imagine that it represents a number that you don't happen to know. At the start of a problem, the value of x is hidden from you. You are normally asked to find the x.
To multiply a known number by a variable, just write the known number in front of the variable. If you want to multiply x by 3, write 3x.
An expression is anything that ultimately represents a number somehow. You might not know that number, but you express it using variables, numbers you know, and operations such as adding, subtracting etc. Within an expression, you have one or more terms. A term involves no addition or subtraction (typically). Often, a term is just a product of variables and known numbers.
5xy + 3x - 2y is an algebraic expression. It has three terms: 1- 5xy, 2- 3x, 3- 2y. In each of the above terms, 5, 3, and 2 are their coefficients.
Value of an expression for a fixed value of an unknown variable can be found by inserting the value of the unknown variable. Example: Find the value of 5a^2 + 7a - 3 when the value of a is 2. Putting 2 in place of a: 5 × 2^2 + 7 × 2 - 3 = 5 × 4 + 14 - 3 = 20 + 14 - 3 = 31.
Like terms are very similar to each other. They only differ by a numerical coefficient. Everything else in them is the same. In 3X^2 + 7X + 2X^2 - 5, the terms 3X^2 and 2X^2 are like terms. Like terms can be combined into one term. Just add or subtract the coefficients. So the expression 3X^2 + 7X + 2X^2 - 5 becomes 3X^2 + 2X^2 + 7X - 5 = 5X^2 + 7X - 5. Whenever a term does not have a coefficient, act as if the coefficient is 1. A negative sign in front of a term on its own can be seen as a -1 coefficient.
Evaluate the following algebraic expressions when a = 3, b = -5, x = 6, y = 12, z = -8.
1. 4a + z = 4(3) - 8 = 12 - 8 = 4
2. 5z^2 - 2z + 2 = 5(-8)^2 - 2(-8) + 2 = 5(64) + 16 + 2 = 320 + 16 + 2 = 338
3. 5xy / 2b = 5(6)(12) / 2(-5) = 5(72) / -10 = -36
4. 8y(a^3 - 2y) = 8(12)[3^3 - 2(12)] = 96[27 - 24] = 96[3] = 288
5. bx(z + 3) = (-5)(6)[-8 + 3] = (-30)(-5) = 150
The distributive property of multiplication tells you how to multiply the terms inside parentheses by the term outside the parentheses.
1. 5a + 2a + 7a = (5 + 2 + 7)a = 14a
2. 4(x + 2y) + 2(x + y) = (4x + 8y) + (2x + 2y) = 6x + 10y
3. 14 + 9(2w + 7) - 2(6 - w) = 14 + (18w + 63) - (12 - 2w) = 65 + 20w
4. 11(4m + 5) + 3(-3m + 8) = 44m + 55 - 9m + 24 = 35m + 79
5. 8(2a - b - 3c) + 3(2a - b) - 4(6 - b) = 16a - 8b - 24c + 6a - 3b - 24 + 4b = 22a - 7b - 24c - 24
To simplify with exponents, don't feel like you have to work only from the rules for exponents. It is often simpler to work directly from the definition and meaning of exponents.
Example: Simplify x^6 × x^5. The x^6 means six copies of x multiplied together, and the x^5 means five copies of x multiplied together. So multiply those two expressions together to get eleven copies of x multiplied together: x^6 × x^5 = x^11.
Example: Simplify 6^8 / 6^5. The 6^8 means eight copies of 6 on top; the 6^5 means five copies of 6 underneath. Cancel five 6's from top and bottom: (6·6·6) / 1 = 6^3.
1. 5x^2 + 8x^2 = 13x^2
2. 5ab^4 - ab^4 = 4ab^4
3. 9mn^3 + 8mn + 2mn^3 = 11mn^3 + 8mn
4. 5c^2 + 3c - 2c^2 + 4 - 7c = 3c^2 - 4c + 4
5. 3x^2 + 4ax - 8a^2 + 7x^2 - 2ax + 7a^2 = 10x^2 + 2ax - a^2
The very simplest case for polynomial multiplication is the product of two one-term polynomials. Example: Simplify (5x^2)(-2x^3).
Two steps are required for multiplying polynomials:
Example: Multiply 3x^2(4x^2 - 5x + 7). Distribute the 3x^2: 3x^2(4x^2 - 5x + 7) = 12x^4 - 15x^3 + 21x^2.
Example: Multiply (3x - 4y)(5x - 2y). Distribute each term and combine like terms: 15x^2 - 6xy - 20xy + 8y^2 = 15x^2 - 26xy + 8y^2.
An equation is a mathematical statement where two expressions are set equal to each other. Using logic and mathematical operations, you can manipulate the equation to find a solution.
Solving equations is not very different from working with numerical or algebraic expressions. If a number is being added to or subtracted from a term on one side of an equation, you can eliminate that number by performing the inverse operation.
1. x - 25 = 32 => x - 25 + 25 = 32 + 25 => x = 57
2. 4x = -20 => 4x / 4 = -20 / 4 => x = -5
3. 3x - 2 = 7 => 3x = 7 + 2 => x = 9 / 3 => x = 3
4. -2x + 6 = 4x - 2 => -6x = -8 => x = 8/6 = 4/3
5. -(-x - 5) + 5(x - 9) = 2(x + 8) - (2x + 5) => x + 5 + 5x - 45 = 2x + 16 - 2x - 5 => 6x - 40 = 11 => 6x = 51 => x = 17/2
The key to solving these types of equations is to move all the terms containing the variable to one, and only one, side. It doesn't matter which side you choose, just try to pick the easier one.
1. 11x + 7 = 3x - 9 => 8x = -16 => x = -2. Check: 11(-2) + 7 = 3(-2) - 9 => -15 = -15.
2. 3x - 23 = 54 - 4x => 7x = 77 => x = 11.
3. 5x + 3 + 6x = 10x + 9 - x => 11x + 3 = 9x + 9 => 2x = 6 => x = 3.
4. 10x + 27 - 5x - 46 = 32 + 3x - 19 => 5x - 19 = 3x + 13 => 2x = 32 => x = 16. Check: 187 - 80 - 46 = 80 - 19 => 61 = 61.
5. 20x - 5 - 5x = 11x + 49 => (as reduced) 8x = 54 => x = 6.75.
6. 0.4 + 3x - 0.25 = 1.15 - 2x => 5x = 1 => x = 1/5.
7. 2x + 17 - 1.2x = 10 - 0.2x + 11 => 0.8x + 17 = 21 - 0.2x => 1x = 4 => x = 4.
8. 2 + 6x - 0.2 = 5x + 2.1 => 6x + 1.8 = 5x + 2.1 => x = 0.3.
9. 1.3 + 5x - 0.1 = -1.2 - 3x => 5x + 0.15 = 1.15 - 2x (as worked) => x = 0.2.
10. 3x + 12 - 0.8x = 3.4 - 0.8x - 9.4 => 2.2x + 12 = -6 - 0.8x => 3x = -18 => x = -6.
Up to now we have solved equations with only one unknown variable. When solving for two unknown variables, two equations are required and these equations are known as simultaneous equations. The solutions are the values of the unknown variables which satisfy both equations simultaneously. In general, if there are n unknown variables, then n independent equations are required to obtain a value for each of the n variables.
We can solve simultaneous equations algebraically using the substitution method or the elimination method.
Example: x + y = -1 ...(1); 3 = y - 2x ...(2). From (1), x = y + 1. Substitute into (2): 3 = y - 2(y + 1) => 3 = -y - 2 => y = -5. Then x = -5 + 1 = -4. Answer: x = -4 and y = -5.
Example: 4y + 3x = 100 ...(1); 4y - 19x = 12 ...(2). From (1), x = (100 - 4y)/3. Substitute into (2): 12y - 19(100 - 4y) = 36 => 88y = 1936 => y = 22. Then x = (100 - 88)/3 = 4. Answer: x = 4 and y = 22.
In the elimination method for solving simultaneous equations, two equations are simplified by adding them or subtracting them. This eliminates one of the variables so that the other variable can be found. To add two equations, add the left hand expressions and right hand expressions separately; to subtract, subtract them separately. Before using the elimination method you may have to multiply every term of one or both of the equations by some number so that equal terms can be eliminated.
Example: 3x + y = 2 ...(1); 6x - y = 25 ...(2). Add (1) and (2): 9x = 27 => x = 3. Then 3(3) + y = 2 => y = -7. Answer: x = 3 and y = -7.
1. x + y = 4; 2x - y = -1. Add: 3x = 3 => x = 1; then 1 + y = 4 => y = 3.
2. 3x = 5 - 7y; 2y = x - 6. Solving gives x = -2 and y = 8.
3. 3x + y = 20; x/3 + 10 = y. Solving gives x = 10 and y = 3 (as worked in the book).
4. 2x + 7y = 45; 3x + 4y = 22. Solving gives x = 5 and y = 0 (as worked in the book).
5. 3x - 5y = -21; 2(2y - x) = 16. Solving gives x = -2 and y = 6.
A Practice Exercise appears here (a numbered set of multiple-choice questions on algebra, expressions and simultaneous equations, with a Directions box and an answer grid at the end). It is available as an interactive quiz in the app, so the individual MCQ options are not reproduced in these notes.
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