Number Sense and Percents Survival Guide Seventh Grade Math Number Sense and Percents

Contents Divisibility Rules Factors and Factor Pairs Multiples Primes and Composites Prime Factorization GCF and GCF Word Problems LCM and LCM Word Problems

Contents Adding Mixed Numbers Subtracting Mixed Numbers Multiplying Mixed Numbers Dividing Mixed Numbers Adding & Subtracting Decimals Multiplying Decimals Dividing Decimals

Contents Changing Fractions to Decimals Changing Decimals to Fractions Changing Fractions to Percents Changing Percents to Fractions Changing Decimals to Percents Changing Percents to Decimals The Percent Proportion

Divisibility Rules 1: All whole numbers are divisible by 1 You can always divide. These rules are shortcuts to determine if one number divides evenly into another number. 1: All whole numbers are divisible by 1 2: All even numbers (numbers ending in 0, 2, 4, 6, or 8) 3: The sum of the digits is divisible by 3 4: The number formed by the last 2 digits are divisible by 4 5: The number in the one’s place is a 0 or 5 6: The number is divisible by 2 AND 3 8: The number formed by the last 3 digits is divisible by 8 9: The sum of the digits is divisible by 9 10: The number in the one’s place is a 0

Divisibility Rules EXAMPLES: 162,720 is divisible by… 1, 2, 3, 4, 5, 6, 8, 9, and 10 1 2 (ends in a 0) 3 (the sum is 1 + 6 + 2 + 7 + 2 + 0 = 18 and 18 ÷ 3 = 6) 4 (20 ÷ 4 = 5) 5 (ends in a 0) 6 (the number is divisible by 2 and 3) 8 (720 ÷ 8 = 90) 9 (sum is 18 and 18 ÷ 9 = 2) 10 (ends in a 0)

Divisibility Rules EXAMPLES: 32,592,165 is divisible by… 1, 3, and 5 1 not 2 (ends in a 5) 3 (the sum is 3+2+5+9+2+1+6+5 = 33 and 33 ÷ 3 = 11) not 4 (4 does not divide evenly into an odd number) 5 (ends in a 5) not 6 (the number is divisible by 3 but not 2) not 8 (8 does not divide evenly into an odd number) not 9 (sum is 33 and 33 ÷ 9 is not a whole number) not 10 (does not end in a 0)

Divisibility Rules EXAMPLES: 745,168 is divisible by… 1, 2, 3, 4, and 8 1 2 (ends in a 8) 3 (the sum is 7+4+5+1+6+8 = 31 and 31 ÷ 3 does not equal a whole number) 4 (68 ÷ 4 = 17) not 5 (does not end in a 0 or 5) not 6 (the number is divisible by 2 but not 3) 8 (168 ÷ 8 = 21) not 9 (sum is 31 and 31 ÷ 9 is not a whole number) not 10 (does not end in a 0)

Factors and Factor Pairs Factors are numbers you multiply together to get a product. 2 and 5 are factors of 10 because 2 x 5 = 10 A number can have many factors. The factors of 12 are 1, 2, 3, 4, 6, and 12 because 1 x 12 = 12, 2 x 6 = 12, and 3 x 4 = 12. Factor pairs are the two numbers used to get the product. There are 3 factor pairs for 18: 1 and 18, 2 and 9, 3 and 6. There are 4 factor pairs for 40: 1 and 40, 2 and 20, 4 and 10, 5 and 8. There is one factor pair for 11: 1 and 11

Multiples The multiples of a number are the product of the number and any other number. If you list the multiples of a number, the list is endless. The first multiple of 3 is 3 because 3 x 1 = 3. The second multiple of 3 is 6 because 3 x 2 = 6. The third multiple of 3 is 9 because 3 x 3 = 9. The fourth multiple of 3 is 12 because 3 x 4 = 12. The tenth multiple of 3 is 30 because 3 x 10 = 30. The seventy-fifth multiple of 3 is 225 because 3 x 75 = 225.

Primes and Composites Number Factors Prime or Composite 1 1 Neither Prime numbers have exactly two factors (1 and itself). Composite numbers have three or more factors. Number Factors Prime or Composite 1 1 Neither 2 1, 2 Prime 3 1, 3 Prime 4 1, 2, 4 Composite 5 1, 5 Prime 6 1, 2, 3, 6 Composite 7 1, 7 Prime 8 1, 2, 4, 8 Composite 9 1, 3, 9 Composite 10 1, 2, 5, 10 Composite 11 1, 11 Prime 12 1, 2, 3, 4, 6, 12 Composite

Primes and Composites Prime numbers are white. Composite numbers are yellow. One is neither prime nor composite!!

Prime Factorization Every number has a unique prime factorization. When a number is written as a product of prime numbers, it is called the prime factorization for that number. The prime factorization for 12 is 2 x 2 x 3 and the prime factorization for 50 is 2 x 5 x 5. Always write the prime numbers from least to greatest. Two common methods for finding the prime factorization is a factor tree and prime division. These two diagrams show how to find the prime factorization for 80 using both common methods.

Greatest Common Factor All natural numbers share a common factor of 1. Many numbers share other factors in common. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The common factors are 1, 2, 3, 4, 6 and 12. The greatest common factor (GCF) is 12. It is the largest factor the numbers have in common. The list method, used above, works well for determining the GCF of smaller numbers. For larger numbers, other methods, such as “the slide” and prime factorization are useful.

Greatest Common Factor The SLIDE Method Prime Factorization Method The GCF can be found The numbers in the prime by multiplying the 2 and factorization that 40 and 72 the 4. The GCF of 40 and have in common are 2 x 2 72 is 8. x 2. The GCF is 8.

Greatest Common Factor The SLIDE Method Prime Factorization Method The GCF can be found The numbers in the prime by multiplying 4 x 3 x 10. factorization that 120 and 360 The GCF of 120 and 360 is have in common are 2 x 2 x 2 120. x 3 x 5 = 120. The GCF is 120.

GCF Word Problems Greatest common factor word problems usually contain one of the following words in the question…greatest, most, largest, maximum, highest. Juliana is putting together first-aid kits. She has 20 large bandages and 8 small bandages, and she wants each kit to be identical, with no bandages left over. What is the greatest number of first-aid kits Juliana could put together? A florist has 5 tulips and 15 carnations. If the florist wants to create identical bouquets without any leftover flowers, what is the maximum number of bouquets the florist can make?

Least Common Multiple All numbers have common multiples. It is often helpful to find the least common multiple (also known as the least common denominator) when adding and subtracting fractions. The first 10 multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60. The first 10 multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80. The common multiples in the list are 24 and 48. The least common multiple (LMC) of 6 and 8 is 24. It is the smallest multiple the two numbers have in common. The list method, used above, works well for determining the LCM of smaller numbers. For larger numbers, other methods, such as “the slide” and prime factorization are useful.

Least Common Multiple The SLIDE Method Prime Factorization Method The LCM can be found To find the LCM, multiply the by multiplying the 2, 4, numbers they have in common 5 and 9. The LCM of 40 (the 2 x 2 x 2) with the numbers and 72 is 360. not in common (3 x 3 x 5). The LCM of 40 and 72 is 360.

Least Common Multiple The SLIDE Method Prime Factorization Method The LCM can be found To find the LCM, multiply the by multiplying the 2, 7, number they have in common and 8. The LCM of 14 (the 2) with the numbers not and 16 is 2 x 7 x 8 = 112. in common (2 x 2 x 2 x 7). The LCM of 14 and 16 is 112.

LCM Word Problems Least common multiple word problems usually contain one of the following words in the question…least, smallest, minimum. Sara's Bath Shop sells bars of soap in boxes of 2 bars and bottles of soap in boxes of 19 bottles. An employee is surprised to discover that the shop sold the same number of bars and bottles last week. What is the smallest number of each type of soap that the shop could have sold? Butler Stationery sells cards in packs of 18 and envelopes in packs of 16. If Evan wants the same number of each, what is the least number of cards that he will have to buy?

LCM Word Problems A second type of least common multiple word problem will have two or more things happening over and over again. The question will want to know when those things will happen again at the same time. Jordan rides his bike every 3 days, swims every 2 days, and runs every 4 days. If he did all three exercises today, in how many days will he do all three again on the same day? Maria cleans her over every 40 days and washes windows every 50 days. If she did both chores today, in how many days will she do both chores again on the same day?

Adding Mixed Numbers Find the least common denominator (LCD). For this problem, the LCD is 20. Rewrite the problem with equivalent fractions using the LCD. Add the numerators. Keep the denominator. Add the whole numbers. Finalize the problem by making sure the fraction is not improper. Always simplify!

Subtracting Mixed Numbers Find the least common denominator (LCD). For this problem, the LCD is 20. Rewrite the problem with equivalent fractions using the LCD. Subtract the numerators. Keep the denominator. Subtract the whole numbers. Always simplify!

Subtracting Mixed Numbers In this problem, you must rename (also known as borrow) before you can subtract the numerators. To rename a mixed number, subtract 1 from the whole number and add it to the fractional part. Change the fractional part to an improper fraction.

Multiplying Mixed Numbers Rewrite every mixed number as an improper fraction. Optional: Cross-cancel and reduce fractions to make the next steps easier! Multiply the numerators straight across. Multiply the denominators straight across. If the answer is improper, change it to a mixed number. Always simplify!

Dividing Mixed Numbers Rewrite every mixed number as an improper fraction. Multiply by the reciprocal. This means change the divide sign to multiply and “flip” the fraction that came after divide sign. Optional: Cross-cancel and reduce fractions to make the next steps easier! Multiply the numerators straight across. Multiply the denominators straight across. If the answer is improper, change it to a mixed number. Always simplify!

Adding & Subtracting Decimals Line up the decimal points! Add zeros after the decimal points, if needed. Bring the decimal point straight down. Add or subtract.

Multiplying Decimals Line up the last digit of each number. There is no need to line up the decimals. Multiply the numbers as if they are whole numbers. Ignore the decimals for now. After multiplying, determine where the decimal should be placed. Count the number of digits after the decimal point for both factors. Your answer needs to have the same number of digits after the decimal point. In this first example, there are three digits after the decimal point (3, 2, 7). As a result, the answer has three digits after the decimal point (0, 2, 4).

Dividing by Whole Numbers When dividing by a whole number, remember to repeat the following steps: divide, multiply, subtract, and bring down the next number. When you reach the end of the whole number, you are no longer to write the leftover number as a remainder! Instead, add a decimal and bring it up to the answer. Then add as many zeros as you need to keep dividing. In some cases, as the second example, you will begin to notice a repeating pattern. Use a bar over the repeating numbers. REMEMBER TO KEEP ALL YOUR NUMBERS AND DECIMALS LINED UP!!

REMEMBER TO KEEP ALL YOUR NUMBERS AND DECIMALS LINED UP!! Dividing by Decimals When dividing by a decimal, you must FIRST make the divisor into a whole number. This is done by moving the decimal to the right. Then count the number of places you moved the decimal point and do the exact same thing to the dividend. The dividend does NOT need to be a whole number! Notice in the first example, when the decimal is moved once in the divisor, it is moved once in the dividend. In the second example, when the decimal is moved twice in the divisor, it is moved twice in the dividend. Zeros are added as needed. REMEMBER TO KEEP ALL YOUR NUMBERS AND DECIMALS LINED UP!!

Changing Fractions to Decimals Many fractions can be changed to decimals by using place value. Here are a few examples: 3 5 is equal to 6 10 . Six tenths is 0.6 as a decimal. 7 25 is equal to 28 100 which is 0.28 as a decimal. When this does not work, remember that a fraction is just another way to write a division problem. 1 8 is equal to 1 ÷ 8 which equals 0.125 7 12 is equal to 7 ÷ 12 which equals 0.58 𝟑

Changing Decimals to Fractions Changing decimals to fractions is as easy as reading the number! Remember to simplify the fraction. 0.32 is thirty-two hundredths = 32 100 = 8 25 0.9 is nine tenths = 9 10 0.012 is twelve thousandths = 12 1000 = 3 250 0.145 is one hundred forty-five thousandths = 145 1000 = 29 200

Changing Decimals to Percents To change a decimal to a percent, move the decimal two places to the right and add a percent sign. 0.57 = 57% 0.29 = 29% 0.5 = 50% 0.06 = 6% 0.002 = 0.2% 1.05 = 105%

Changing Percents to Decimals To change a percent to a decimal, move the decimal two places to the left and remove the percent sign. 36% = 0.36 45% = 0.45 70% = 0.70 = 0.7 8% = 0.8 1.9% = 0.019 435% = 4.35

Changing Percents to Fractions Percent means per hundred. Every percent can be changed into a fraction by placing the percent over 100. Remember to simplify. Examples: 68% = 68 100 = 17 25 115% = 115 100 = 1 15 100 = 1 3 20 If the percent contains a decimal, a little more work is required. Change the percent to a decimal and then to a fraction. Examples: 37.5% = 0.375 = 375 1000 = 3 8 Examples: 5.25% = 0.0525 = 525 10000 = 21 400

Changing Fractions to Percents Percent means per hundred. If you can write an equivalent fraction with a denominator of 100, the numerator is the percent. Examples: 12 25 = 48 100 so 12 25 = 48% 36 300 = 12 100 so 36 300 = 12% When this doesn’t work, change the fraction to a decimal by dividing and then change the decimal to a percent. Examples: 19 32 = 19 ÷ 32 = 0.59375 = 59.375%

The Percent Proportion is is of = % 100 . Percent Problems: The % Proportion The Percent Proportion is is of = % 100 . Examples: What percent of 50 is 30? 30 50 = % 100 Answer: 60% 32 is 40% of what number? 32 x = 40 100 Answer: 80 75% of 420 is what number? x 420 = 75 100 Answer: 315 115% of what number is 805? 805 x = 115 100 Answer: 700

Percent Problems: The % Proportion For story problems, this percent proportion works better: part whole = % 100 Example 1: 56% of the students in a class wear flip-flops. If there are 32 students, how many students were flip-flops? Since 32 students represents the whole class, the proportion would be x 32 = 56 100 . Solve to find18 students wear flip-flops. Example 2: On a quiz, Marcy got 38 questions correct out of 40. What percent did Marcy get on the test? The proportion 38 40 = 𝑥 100 would give you an answer of 95%.