NTS Pattern And Practice: Quantitative Ability

Quantitative Ability

The Quantitative section measures your basic mathematical skills, understanding of elementary mathematical concepts, and the ability to reason quantitatively and solve problems in a quantitative setting. There is a balance of questions requiring basic knowledge of arithmetic, algebra, geometry, and data analysis. These are essential content areas usually studied at the high school level.

The questions in the quantitative section can also be from

• Discrete Quantitative Question
• Quantitative Comparison Question
• Data Interpretation Question etc.

The distribution in this guide is only to facilitate the candidates. This distribution is not a part of test template, so, a test may contain all the questions of one format or may have a random number of questions of different formats.

This chapter is divided into 4 major sections. The first discusses the syllabus/contents in each section of the test respectively and the remaining three sections address the question format, guide lines to attempt the questions in each format and some example questions.

General Mathematics Review

Arithmetic

The following are some key points, which are phrased here to refresh your knowledge of basic arithmetic principles.

Basic arithmetic

• For any number a, exactly one of the following is true:
• a is negative
• a is zero
• a is positive
• The only number that is equal to its opposite is 0 (e.g. a = −a only if a = 0 ).
• If 0 is multiplied to any other number, it will make it zero ( a×0 = 0 ).
• Product or quotient of two numbers of the same sign are always positive and of a different sign are always negative. E.g. if a positive number is multiplied to a negative number the result will be negative and if a negative number is divided by another negative number the result will be positive.

See the following tables for all combinations.

+	÷ or ×	+	=	+
+	÷ or ×	-	=	-
-	÷ or ×	+	=	-
-	÷ or ×	-	=	+

• The sum of two positive numbers is always positive.
• The sum of two negative numbers is always negative.
• Subtracting a number from another is the same as adding its opposite
  a − b = a + (−b)
• The reciprocal of a number a is 1/a.
• The product of a number and its reciprocal is always one a x 1/a = 1.
• Dividing by a number is the same as multiplying by its reciprocal a ÷ b = a x 1/b.
• Every integer has a finite set of factors (divisors) and an infinite set of multipliers.
• If a and b are two integers, the following four terms are synonyms.
• a is a divisor of b
• a is a factor of b
• b is a divisible by a
• b is a multiple of a

They all mean that when a is divided by b there is no remainder.

• Positive integers, other than 1, have at least two positive factors.
• Positive integers, other than 1, which have exactly two factors, are known as prime numbers.
• Every integer greater than 1 that is not a prime can be written as a product of primes.
  To find the prime factorization of an integer, find any two factors of that number, if both are primes, you are done; if not, continue factorization until each factor is a prime.
  E.g. to find the prime factorization of 48, two factors are 8 and 6. Both of them are not prime numbers, so continue to factor them.
  Factors of 8 are 4 and 2, and of 4 are 2 and 2 (2 × 2 × 2).
  Factors of 6 are 3 and 2 (3 × 2).
  So the number 48 can be written as 2 × 2 × 2 × 2 × 3.
• The Least Common Multiple (LCM) of two integers a and b is the smallest integer which is divisible by both a and b, e.g. the LCM of 6 and 9 is 18.
• The Greatest Common Divisor (GCD) of two integers a and b is the largest integer which divides both a and b, e.g. the GCD of 6 and 9 is 3.
• The product of GCD and LCM of two integers is equal to the products of numbers itself. E.g.
  6 x 9 = 54
  3 x 8 = 54 (where 3 is GCD and 18 is LCM of 6 and 9).
• Even numbers are all the multiples of 2 e.g. (… , −4, −2, 0, 2, 4, …).
• Odd numbers are all integers not divisible by 2 ( … , −5, −3, −1, 1, 3, 5, … ).
• If two integers are both even or both odd, their sum and difference are even.
• If one integer is even and the other is odd, their sum and difference are odd.
• The product of two integers is even unless both of them are odd.
• When an equation involves more than one operation, it is important to carry them out in the correct order. The correct order is Parentheses, Exponents, Multiplication and Division, Addition and Subtraction, or just the first letters PEMDAS to remember the proper order.

Exponents and Roots

• Repeated addition of the same number is indicated by multiplication:
  17 + 17 + 17 + 17 + 17 = 5 × 17
• Repeated multiplication of the same number is indicated by an exponent:
  17 × 17 × 17 × 17 × 17 = 17⁵

In the expression 17⁵, 17 is called base and 5 is the exponent.

• For any number b: b¹ = b and bⁿ = b × b × … × b, where b is used n times as factor.
• For any numbers b and c and positive integers m and n:
• b^m x bⁿ = b^{m + n}
• b^m ÷ bⁿ = b^{m - n}
• (b^m)ⁿ = b^{m x n}
• b^m x c^m = (bc)^m
• If a is negative, aⁿ is positive if n is even, and negative if n is odd.
• There are two numbers that satisfy the equation x² = 9: x = 3 and x = −3. The positive one, 3, is called the (principal) square root of 9 and is denoted by √9 . Clearly, each perfect square has a square root:
  √0 = 0 , √9 = 3, √36 = 6 , √169 = 13 , √225 = 25 etc.
• For any positive number a there is a positive number b that satisfies the
  equation √a = b .
• For any positive integer, ( √a)² = √a × √a = a .
• For any positive numbers a and b:
• √ab = √a × √b and √a/b = √a/√b
• √a+b != √a + √b
• as 5 = √25 = √9+16 != √9 + √16 = 3 + 4 = 7
• Although it is always true that (√a)² = a , √a² = a is true only if a is
  positive as √(-5)² = √25 = 5 != -5
• For any number a, √aⁿ = a^n/2
• For any number a, b, and c:
  
• a(b + c) = ab + ac, a(b − c) = ab − ac and if a != 0
• (b + c) / a = b / a + c / a, (b - c) / a = b / a - c / a

Inequalities

• For any number a and b, exactly one of the following is true: a > b or a = b or a < b.
• For any number a and b, a > b means that a − b is positive.
• For any number a and b, a = b means that a − b is zero.
• The symbol ≥ means greater than or equal to and the symbol ≤ means less than or equal to. E.g. the statement x ≥ 5 means that x can be 5 or any number greater than 5.The statement 2 < x < 5 is an abbreviation of 2 < x and x < 5.
• Adding or subtracting a number to an inequality preserves it.
• If a < b , then a + c < b + c and a − c < b − c.
  e.g. 5 < 6⇒5 +10 < 6 +10 and 5 −10 < 6 −10
• Adding inequalities in same direction preserves it:
  If a < b and c < d , then a + c < b + d .
• Multiplying or dividing an inequality by a positive number preserves it. If
  a < b and c is a positive number, then a × c < b × c and a/c < b/c.
• Multiplying or dividing an inequality by a negative number reverses it. If a < b and c is a negative number, then a × c > b × c and a/c > b/c
• If sides of an inequality are both positive and both negative, taking the reciprocal reverses the inequality.
• If 0 < x <1 and a is positive, then xa < a.
• If 0 < x < 1 and m and n are integers with m > n , then x^m < xⁿ < x.
• If 0 < x <1, then √x > x.
• If 0 < x <1, then 1/x > x and 1/x > 1.

Properties of Zero

• 0 is the only number that is neither negative nor positive.
• 0 is smaller than every positive number and greater than every negative number.
• 0 is an even integer.
• 0 is a multiple of every integer.
• For every number a : a + 0 = a and a − 0 = a.
• For every number a : a×0 = 0.
• For every posi tive integer n : 0ⁿ = 0.
• For every number a (including 0) : a ÷ 0 and a / 0 are undefined symbols.
• For every number a (other than 0) : 0 ÷ a = 0 /a = 0.
• 0 is the only number that is equal to i ts opposite : 0 = −0.
• If the product of two or more numbers is 0, at least one of them is 0.

Properties of One

• For any number a : a×1= a and a/1 = a.
• For any number n : 1ⁿ = 1.
• 1 is the divisor of every integer.
• 1 is the smallest positive integer.
• 1 is an odd integer.
• 1 is not a prime.

Fractions and Decimals

• When a whole is divided into n equal parts, each part is called one nth of the whole, written 1/n. For example, if a pizza is cut (divided) into 8 equal slices, each slice is one eighth (1/8) of the pizza; a day is divided into 24 equal hours, so an hour is one twenty-fourth (1/24) of a day and an inch is one twelfth (1/12 ) of a foot. If one works for 8 hours a day, he works eight twenty-fourth (8/24 ) of a day. If a hockey stick is 40 inches long, it measures forty twelfths (40/12) of a foot.
• The numbers such as 1/8 , 1/24 , 8/24 and 40/12 , in which one integer is written over the second integer, are called fractions. The center line is called the fraction bar. The number above the bar is called the numerator, and the number below the bar is called denominator.
• The denominator of a fraction can never be 0.
• A fraction, such as 1/24 , in which the denominator is greater than numerator, is known as a proper fraction. Its value is less than one.
• A fraction, such as 40/12 , in which the denominator is less than numerator, is known as an improper fraction. Its value is greater than one.
• A fraction, such as, 12/12 in which the denominator is equal to the numerator, is also known as an improper fraction. But, Its value is one.
• Every fraction can be expressed in decimal form (or as a whole number) by dividing the number by the denominator.
  3/10 = 0.3, 3/4 = 0.75, 8/8 = 1, 48/16 = 3, 100/8 = 12.5.
• Unlike the examples above, when most fractions are converted to decimals, the division does not terminate, after 2 or 3 or 4 decimal places; rather it goes on forever with some set of digits repeating it.
  2/3 = 0.66666..., 3/11 = 0.272727..., 5/12 = 0.416666..., 17/15 = 1.133333...

• To compare two decimals, follow these rules:
• Whichever number has the greater number to the left of the decimal point is greater: since 11 > 9, 11.0001 > 9.8965 and since 1 > 0, 1.234 > .8. (Recall that if a decimal is written without a number on left of decimal point, you may assume that a 0 is there, so, .8 = 0.8).
• If the numbers to the left of the decimal point are equal, proceed as follows:
• If the numbers do not have the same number of digits to the right of the decimal point, add zeroes to the end of the shorter one to make them equal in length.
• Now compare the numbers ignoring the decimal point.
• For example, to compare 1.83 and 1.823, add a 0 to the end of 1.83 forming 1.830. Now compare them, thinking of them as whole numbers without decimal point: since 1830 > 1823, then 1.830 >1.823.

• There are two ways to compare fractions:
• Convert them to decimals by dividing, and use the method already described to compare these decimals. For example to compare 2/5 and 1/4 , convert them to decimals. 2/5 = 0.4 and 1/4 = 0.25. Now, as 0.4 > 0.25, 2/5 > 1/4.
• Cross multiply the fractions. For example to compare 2/5 and 1/4, cross multiply: Since 2×4>1×5, then 2/5 > 1/4.
• While comparing the fractions, if they have same the denominators, the fraction with the larger numerator is greater. For example 3/5 > 2/5.
• If the fractions have the same numerator, the fraction with the smaller denominator is greater. For example 3/5 > 3/10.
• Two fractions are called equivalent fractions if both of them have same decimal value.
• For example, 1/2 = 5/10 as both of these are equal to 0.5.
• Another way to check the equivalence of two fractions is to cross-multiply. If both of the products are same, the fractions are equivalent. For Example, to compare 2/5 with 6/15, cross-multiply. Since 2×15 = 6×5 , both of the fractions are equivalent.
• Every fraction can be reduced to lowest terms by dividing the numerator and denominator by their greatest common divisor (GCD). If the GCD is 1, the fraction is already in lowest terms. For example to reduce 10/15 , divide both numerator and denominator by 5 (which is GCD of 10 and 15). This will reduce the fraction to 2/3.
• To multiply two fractions, multiply their numerators and multiply their denominators. For example 3/5 x 4/7 = 3 x 4 / 5 x 7 = 12/35.
• To multiply a number to a fraction, write that number as a fraction whose denominator is 1. For example 3/5 x 7 = 3/5 x 7/1 = 3 x 7 / 5 x 1 = 21/5.
• When a problem requires you to find the fraction of a number, multiply that fraction with the number. For example,
  to find two fifth (2/5 ) of 200, multiply: 2/5 x 200 = 2/5 x 200/1 = 400/5 = 80.
• The reciprocal of a fraction a/b is another fraction b/a since a/b x b/a = 1.
• To divide one fraction by the other fraction, multiply the reciprocal of divisor with the dividend. For example, 22/7 ÷ 11/7 = 22/7 x 7/11 = 2/1 = 2.
• To add or subtract the fractions with same denominator, add or subtract numerators and keep the denominator. For example 4/9 + 1/9 = 5/9 and 4/9 - 1/9 = 3/9.

Percents

• The word percent means hundredth. We use the symbol % to express the word percent. For example “15 percent” means “15 hundredths” and can be written with a % symbol, as a fraction, or as a decimal. 20% = 20/100 = 0.20.
• To convert a decimal to a percent, move the decimal point two places to the right, adding 0s is necessary, and add the percent symbol (%). For example, 0.375 = 37.5% 0.3 = 30% 1.25 = 125% 10=1000%
• To convert a fraction to a percent, first convert that fraction to decimal, than use the method stated above to convert it to a percent.
• To convert a percent to a decimal, move the decimal point two places to the left and remove the % symbol. Add 0s if necessary. For example, 25% = 0.25 1% =0.01 100% = 1
• You should be familiar with the following basic conversions: 1/2 = 5/10 = 0.50 = 50%, 1/5 = 2/10 = 0.20 = 20%
  1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%
• For any positive integer a : a% of 100 is a.
• For any positive numbers a and b : a% of b = b% of a
• The percent change in the quantity is (actual change)/(original amount) x 100%. For example:
  If the price of a lamp goes from Rs.80 to Rs.100, the actual increase is Rs.20, and the percent increase is 20/80 x 100% = 1/4 x 100% = 25%.
• If a < b , the percent increase in going from a to b is always greater than percent decrease in going from b to a.
• To increase a number by k%, multiply it by 1 + k%, and to decrease a number by k%, multiply it by 1 − k%. For example, the value of an investment of Rs. 20,000 after 25% increase is 20,000 × (1 + 25%) = 20,000 ×( 1.25) = 25,000.
• If a number is the result of increasing another number by k%, to find the original number divide i t by 1 + k%, and if a number is the result of decreasing another number by k%, to find the original number, divide it by 1 − k%.
  For example, The government announced an 20% increase in salaries. If after the increment, The salary of a particular employee is Rs. 18, 000, what was the original salary?
  Original Salary (in Rs.) = Current Salary / (1 + percent increase) = 18,000/(1+20%) = 18,000/1.20 = 15,000.

Ratios and Proportions

• A ratio is a fraction that compares two quantities that are measured in the same units. The first quantity is the numerator and the second quantity is denominator. For example, if there are 16 boys and 4 girls, we say that the ratio of the number of boys to the number of girls on the team is 16 to 4, or 16/4. This is often written as 16:4. Since a ratio is just a fraction, it can be reduced or converted to a decimal or a percent. The Following are different ways to express the same ratio: 16 to 4, 16:4, 16/4, 4/1, 0.25, 25%.
• If a set of objects is divided into two groups in the ration a : b , then the first group contains a/(a+b) of the total objects and similarly the second group contains b/(b+c) of the total number of objects. This rule applies to extended ratios, as well. If a set is divided into three groups in the ratio a : b : c, then the first group contains a/(a+b+c) of the total objects, and so on.
• A proportion is an equation that states that two ratios are equivalent. Since ratios are just fractions, any equation such as 4/6 = 10/15 in which each side is a single fraction is proportion. This proportion states that 4 relates to 6 in same ratio as 10 relates to 15.
• For each proportion of the form a/b = c/d, ad = bc. This property can be used to solve proportions for unknowns (variables). For example: “If 3 oranges cost Rs.5, how many oranges can you buy for Rs.100”. To solve this problem we have to set up a proportion. If the number of oranges for Rs.100 is x , then:
  3/5 = x/100 = 3 x 100 = x multiply 5 ⇒ x = (3 x 100) / 5 ⇒ x = 60.

Averages

• The average of a set of n numbers is the sum of those numbers divided by n.
  average = (sum of n numbers) / n or simply A = Sum / n.
  the technical name for these kind of averages is Arithmetic Mean.
• If you know the average of n numbers, multiply that average with n to get the sum of numbers.
• If all the numbers in a set are the same, then that number is the average.

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