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This chapter is from the book

This chapter is from the book

Arithmetic

This book assumes a basic understanding of arithmetic. Our focus will be on reviewing some general concepts and applying those concepts to questions that might appear on the PSAT math sections.

Understanding Operations Using Whole Numbers, Decimals, and Fractions

Following are some simple rules to keep in mind regarding whole numbers, fractions, and decimals:

  1. Ordering is the process of arranging numbers from smallest to greatest or from greatest to smallest. The symbol > is used to represent "greater than," and the symbol < is used to represent "less than." To represent "greater than and equal to," use the symbol ≥; to represent "less than and equal to," use the symbol ≤.

  2. The Commutative Property of Multiplication can be expressed as a x b = b x a, or ab = ba. When two numbers are multiplied together, the order they are in doesn’t matter. For example: 2 x 3 = 6, and 3 x 2 = 6.

  3. The Distributive Property of Multiplication can be expressed as a(b + c) = ab + ac. Multiply each element within the parentheses by the element outside of the parentheses. For example: 2(2x + 4) = 4x + 8.

  4. The Associative Property of Multiplication can be expressed as (a x b) x c = a x (b x c). It doesn’t matter in which order the numbers are multiplied. For example: (2 x 3) x 4 = 6 x 4, or 24; 2 x (3 x 4) = 2 x 12, or 24.

  5. The Order of Operations for whole numbers can be remembered by using the acronym PEMDAS:

    • P: First, do the operations within the parentheses, if any.

      E: Next, do the exponents, if any.

      M: Next, do the multiplication, in order from left to right.

      D: Next, do the division, in order from left to right.

      A: Next, do the addition, in order from left to right.

      S: Finally, do the subtraction, in order from left to right.

  6. When a number is expressed as the product of two or more numbers, it is in factored form. Factors are all the numbers that will divide evenly into one number. For example, 1, 3, and 9 are factors of 9: 9 ÷ 1 = 9; 9 ÷ 3 = 3; 9 ÷ 9 = 1.

  7. A number is called a multiple of another number if it can be expressed as the product of that number and a second number. For example, the multiples of 4 are 4, 8, 12, 16, and so on, because 4 x 1 = 4, 4 x 2 = 8, 4 x 3 = 12, 4 x 4 = 16, and so on.

  8. The Greatest Common Factor (GCF) is the largest number that will divide evenly into any two or more numbers. The Least Common Multiple (LCM) is the smallest number that any two or more numbers will divide evenly into. For example, the GCF of 24 and 36 is 12 because 12 is the largest number that will divide evenly into both 24 and 36. The LCM of 24 and 36 is 72 because 72 is the smallest number that both 24 and 36 will divide evenly into.

  9. Multiplying and dividing both the numerator and the denominator of a fraction by the same non-zero number will result in an equivalent fraction. For example, , which can be reduced to . This works because any non-zero number divided by itself is always equal to 1.

  10. When multiplying fractions, multiply the numerators to get the numerator of the product, and multiply the denominators to get the denominator of the product. For example, .

  11. To divide fractions, multiply the first fraction by the reciprocal of the second fraction. In mathematics, the reciprocal of any number is the number that, when multiplied by the first number, will yield a product of 1. For example: The reciprocal of is because . So, , which equals .

  12. When adding and subtracting like fractions (fractions with the same denominator), add or subtract the numerators and write the sum or difference over the denominator. So, , and .

  13. When adding or subtracting unlike fractions, first find the Lowest (or Least) Common Denominator (LCD). The LCD is the smallest number that all the denominators will divide evenly into. For example, to add and , find the smallest number that both 4 and 6 will divide evenly into. That number is 12, so the LCD of 4 and 6 is 12. Multiply by to get , and multiply by to get . Now you can add the fractions, as follows: , which can be simplified to .

  14. To convert a mixed number into an improper fraction, first multiply the whole number by the denominator, add the result to the numerator, and place that quantity over the denominator. For example, , or . To convert an improper fraction into a mixed number, first determine how many times the denominator will divide evenly into the numerator. Then, place the remainder over the denominator, as follows: In the improper fraction , 8 divides evenly into 21 twice (8 x 2 = 16), with a remainder of 5 (21 – 16 = 5). Therefore, , is equal to .

  15. When converting a fraction to a decimal, divide the numerator by the denominator. For example , = 3 ÷ 4, or .75.

  16. Place value refers to the value of a digit in a number relative to its position. Starting from the left of the decimal point, the values of the digits are ones, tens, hundreds, and so on. Starting to the right of the decimal point, the values of the digits are tenths, hundredths, thousandths, and so on.

Understanding Squares and Square Roots

In mathematics, a square is the product of any number multiplied by itself, and is expressed as a2 = n. A square root is written as , and is the non-negative value a that fulfills the expression a2 = n. For example, the square root of 25 would be written as . The square root of 25 is 5, and 5-squared, or 52, equals 5 x 5, or 25. A number is considered a perfect square when the square root of that number is a whole number. So, 25 is a perfect square because the square root of 25 is 5, which is a whole number.

When you square a fraction, simply calculate the square of both the numerator and the denominator.

  • =

Understanding Exponents

When a whole number is multiplied by itself, the number of times it is multiplied is referred to as the exponent. As shown above with square roots, the exponent of 52 is 2 and it signifies 5 x 5. Any number can be raised to any exponential value.

  • 76 = 7 x 7 x 7 x 7 x 7 x 7 = 117,649.

Remember that when you multiply a negative number by a negative number, the result will be a positive number. When you multiply a negative number by a positive number, the result will be a negative number. These rules should be applied when working with exponents, too.

  • -32 = -3 x -3

  • -3 x -3 = 9

  • -33 = -3 x -3 x -3

  • (–3 x –3) = 9

  • 9 x –3 = -27

The basic rules of exponents follow:

  • am x an = a(m+n)

  • (am)n = amn

  • (ab)m = am x bm

  • a0 = 1, when a ≠ 0

  • , when a ≠ 0

  • , when b ≠ 0

Understanding Ratios and Proportions

A ratio is the relationship between two quantities expressed as one divided by the other.

For example: Kate works 2 hours for every 3 hours that Darleen works. This can be expressed as or 2:3, and is known as a part-to-part ratio. If you compared the number of hours that Kate works in one week to the total number of hours that every employee works in one week, that would be a part-to-whole ratio.

A proportion is an equation in which two ratios are set equal to each other.

For example: Kate worked 30 hours in one week and earned $480. If she received the same hourly rate the next week, how much would she earn for working 25 hours that week?

  • ;solve for x

  • 30x = 12,000

  • x = 400

Kate would earn $400 that week.

Understanding Percent and Percentages

A percent is a fraction whose denominator is 100. The fraction is equal to 55%.

Percentage problems often deal with calculating an increase or a decrease in number or price.

For example: A jacket that originally sells for $90 is on sale for 35% off. What is the sale price of the jacket (not including tax)?

  • 35% of $90 =

  • .35 x $90 = $31.50

The discount is equal to $31.50, but the question asked for the sale price. Therefore, you must subtract $31.50 from $90.

  • $90.00 – $31.50 = $58.50

You could also more quickly solve a problem like this by recognizing that, if the jacket is 35% off of the regular price, the sale price must be equal to 100% – 35%, or 65% of the original price.

  • 65% of $90 =

  • .65 x $90 = $58.50.

Understanding Simple Probability

Probability is used to measure how likely an event is to occur. It is always between 0 and 1; an event that will definitely not occur has a probability of 0, whereas an event that will certainly occur has a probability of 1. To determine probability, divide the number of outcomes that fit the conditions of an event by the total number of outcomes. For example, the chance of getting heads when flipping a coin is 1 out of 2, or . There are 2 possible outcomes (heads or tails) but only 1 outcome (heads) that fits the conditions of the event. Therefore, the probability of the coin toss resulting in heads is or .5.

When two events are independent, meaning the outcome of one event does not affect the other, you can calculate the probability of both occurring by multiplying the probabilities of each of the events together. For example, the

probability of flipping 3 heads in a row would be x x , or .

Understanding Number Lines and Sequences

A number line is a geometric representation of the relationships between numbers, including integers, fractions, and decimals. The numbers on a number line always increase as you move to the right and decrease when you move to the left. Number line questions typically require you to determine the relationships among certain numbers on the line.

An arithmetic sequence is one in which the difference between consecutive terms is the same. For example, 2, 4, 6, 8..., is an arithmetic sequence where 2 is the constant difference. In an arithmetic sequence, the nth term can be found using the formula an = a1 + (n –1)d, where d is the common difference. A geometric sequence is one in which the ratio between two terms is constant. For example, , 1, 2, 4, 8..., is a geometric sequence where 2 is the constant ratio. With geometric sequences, you can find the nth term using the formula an = a1(r)n – 1, where r is the constant ratio.

Typically, if you can identify the pattern or the relationship between the numbers, you will be able to answer the question. Following is an example of a sequence question similar to one you might find on the PSAT:

    0,1,2,0,1,2,...

The numbers 0, 1, and 2 repeat in a sequence, as shown above. If this pattern continues, what will be the sum of the ninth and twelfth numbers in the sequence?

  • To solve this problem, simply recognize that the third number in the sequence is 2, which means that both the ninth number and the twelfth number in the sequence will also be 2. Therefore, the sum of the ninth and twelfth numbers is 4.

Understanding Absolute Value

The absolute value of a number is indicated by placing that number inside two vertical lines. For example, the absolute value of 10 is written as follows: | 10 |. Absolute value can be defined as the numerical value of a real number without regard to its sign. This means that the absolute value of 10, | 10 |, is the same as the absolute value of -10, | -10 |, in that they both equal 10. Think of it as the distance from -10 to 0 on the number line, and the distance from 0 to 10 on the number line...both distances equal 10 units, as shown below:

Following is an example of how to solve an absolute value question:

  • |3 – 5| =

  • |–2| = 2

Understanding Mean, Median, and Mode

The arithmetic mean refers to the average of a set of values. For example, if a student received grades of 80%, 90%, and 95% on three tests, the average test grade is 80 + 90 + 95 ÷ 3. The median is the middle value of an ordered list. If the list contains an even number of values, the median is simply the average of the two middle values. It is important to put the values in either ascending or descending order before selecting the median. The mode is the value or values that appear the greatest number of times in a list of values.

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