Math End-Term 1 - Lesson with Answers 2024 PDF

End-Term Math Review  Agenda Review: Integers Factors, multiples and prime numbers Fractions, decimals and percentages Ratios Percentage change Units of conversions Powers and roots Integers  Integers Integers are a set of whole numbers i...

End-Term Math Review  Agenda Review: Integers Factors, multiples and prime numbers Fractions, decimals and percentages Ratios Percentage change Units of conversions Powers and roots Integers  Integers Integers are a set of whole numbers including positive numbers, negative numbers and zero. negatives Numbers below zero (left of 0) are called ___________. positives Numbers above zero (right of 0) are called ___________. - 35 What is the opposite of 35? 17 What is absolute value of -17? - 95 What is the opposite of 95?  Integers Select the integers - 35/3 17 95 35.86 17.5 - 95 35 17,000 - 95.99 Integers are whole numbers and their opposites  Integer Rules: + and - same sign. If the signs are the same, the answer has the _____ + + + = ? + - + - = ?- If the signs are different, the answer will take the sign of the _____ larger number. + + - = ?- - + + = ?+ When subtracting integers, change the subtraction to an ________ addition and change the second number to it's _________. opposite Now you can use the addition rules!  Integer Rules: x and ÷ positive If the signs are the same answer will be ________. negative If the signs are different the answer will be ________. Multiplication Division + × + = ?+ + ÷ + = ?+ - × + = ?- - ÷ + = ?- + × - = ?- + ÷ - = ?- - × - = ?+ - ÷ - = ?+  Integer Rules 1) Solve: -56 - (-9) = -47 2) Solve: -8 - (-36) = 28 3) The sum of negative three and negative -8 five is ____________. 4) Solve: 29 + (-7) = 22 5) Solve: -9 x (-8) = 72 Multiples, Factors and Prime Numbers  Lowest Common Multiples (LCM) smallest The LCM (lowest common multiple) is the ________ shared multiple that is _______. What are the What are the Multiples of 3: Multiples of 4: LCMisofthe What 3 &LCM 4: first 5 multiples first 5 multiples 3, 6, 9, 12, 15 4, 8, 12, 16, 20 of 3 12 and 4? of 3? of 4? 6) Find the LCM of 8 and 12. 8, 16, 24, 32, 40... Multiples of 8: __________________ 12, 24, 36, 48, 60... Multiples of 12: __________________ 24 LCM = ____  Lowest Common Multiples (LCM) 7) Find the Lowest Common Multiple (LCM) of 3 and 8 3, 6, 9, 12, 15, 18, 21, 24... Multiples of 3: ______________________ 8, 16, 24, 32, 40... Multiples of 8: __________________ 24 LCM = ____ 8) Find the Lowest Common Multiple (LCM) of 7 and 8 7, 14, 21, 28, 35, 42, 49, 56... Multiples of 7: _________________________ 8, 16, 24, 32, 40, 48, 56, 64... Multiples of 8: _________________________ 56 LCM = ____  Highest Common Factors (HCF) largest The HCF (highest common factor) is the ________ factor that is _________. shared Factors What areof 8 the Factors What areofthe12 Whatofis8the HCF andHCF 12: 1, 2, 4,8? factors 8 1, 2, 3, 4, factors of 6, 12?12 4 12? of 8 and 9) Find the HCF of 24 and 36. 1, 2, 3, 4, 6, 8, 12, 24 Factors of 24: ____________________ 1, 2, 3, 4, 6, 9, 12, 18, 36 Factors of 36: _______________________ 12 HCF = ____  Highest Common Factor (HCF) 10) Find the Highest Common Factor (HCF) of 60 & 15. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 Factors of 60: _______________________________ 1, 3, 5, 15 Factors of 15: ______________________ 15 HCF = ____ 11) Find the Highest Common Factor (HCF) of 50 & 30. 1, 2, 5, 10, 25, 50 Factors of 50: _________________________ 1, 2, 3, 5, 6, 10, 15, 30 Factors of 30: _________________________ 10 HCF = ____  Prime Numbers Prime numbers are any number that is greater than 1 one and _____. and has only 2 factors - ____ itself 2, 3, 5, 7, 11, 13, 17, 19, 23 - List the first 9 prime numbers:_______________________ - Prime factors are prime numbers that ______ divide a number. - Use a _________ factor tree to find the prime factors of a number. Find the prime factors: x 5 x125 5x5 2 x 30 3x5 x  Prime Numbers Which of the following numbers are NOT a prime number? 13 17 15 9 23 14 8 7 91 Fractions  Fractions Fractions are numbers that represent a part of the whole. A class has 30 students. 18 are girls. Write as a fraction: 18 3 ÷ 6 The number numerator on the top is the _________. = 30 5 ÷ 6 The number on the bottom is the denominator ___________. divide the numerator and To simplify the fraction, ______ denominator by the HCF.  Fraction Rules: + and - To add or subtract fractions: numerators and keep the Add/subtract the __________ denominators the _____. same If the denominators are different, you will need to common denominator FIRST. find a ________ 5 2 3 1 1 2 5 8 13 - =6 = + = 20 + 20 = 20 6 6 2 4 5 Remember to multiply the numerator and denominator by the same number!  Fraction Rules: + and - 12) Solve your answer in its simplest form: 8 7 40 21 61 3 + 5 = 15 + 15 = 15 13) Solve your answer in its simplest form: 2 3 8 18 26 13 + = + = = 6 4 24 24 24 12  Fraction Rules: x and ÷ To multiply fractions, multiply the 3 1 3 numerators together and multiply x = the denominators together. 4 5 20 reciprocal the second fraction and To divide, first __________ multiply Now you can change the divide to ________. multiply the fractions together! Don't forget to 2 4 2 5 10 5 simplify your ÷ = x = = answer! 3 5 3 4 12 6  Fraction Rules: x and ÷ 14) Solve your answer in its simplest form: 6 7 42 7 9 x 8 = 72 = 12 15) Solve your answer in its simplest form: 2 3 6 1 6 x 4 = = 4 24  Fraction Rules: x and ÷ 16) Solve your answer in its simplest form: 2 5 2 3 6 2 3 ÷ 3 = 3 x 5 = = 15 5 17 Solve your answer in its simplest form: 3 7 3 4 12 3 4 ÷ 4 = 4 x 7 = = 28 7  Converting Fractions, Decimals and Percentages We can convert between fractions (1/2), decimals (0.5) and percentages (50%) without changing the value. divide the numerator To convert a fraction to decimal, ______ by denominator. multiply by 100. To convert a decimal to a percentage, _______ divide by 100. To convert a percentage to a decimal, _______ Convert the following numbers: to percentage: to decimal: to decimal: 35% 1.25 0.4 0.35 125% 2/5 18) What is 0.80 as a fraction in its simplest form? 0.80 8 4 = = 1 10 5 19) What is 0.65 as a fraction in its simplest form? 0.65 65 13 = = 1 100 20 20) A shop sells 65 markers. Five of them are red. What percentage are not red? 60 65 - 5 = 60 = 0.92 = 92% 65 21) A shop sells 45 donuts. ten of them are chocolate flavored. What percentage are not chocolate flavored? 35 45 - 10 = 35 = 0.78 = 78% 45 22) If 70% of the students are boys, then what fraction of students are girls? 100% - 70% = 30% = 0.3 0.3 3 = 1 10 23) If 42% of the students are boys, then what fraction of students are girls? 100% - 42% = 58% = 0.58 0.58 58 29 = = 1 100 50 24) Convert 3/7 into a decimal. Round to two decimal places. 3 ÷ 7 = 0.4285 = 0.43 25) Convert 5/6 into a decimal. Round to two decimal places. 5 ÷ 6 = 0.8333 = 0.83 26) What is 0.65 as a percentage? 0.65 x 100 = 65 = 65% 27) What is 0.23 as a percentage? 0.23 x 100 = 23 = 23% Ratios  Ratios A ratio compares the sizes of parts or quantities to each other. What is the ratio of red to yellow to blue? red : yellow : blue Simplify using 12 : 4 : 8 the HCF of all 3 : 1 : 2 the numbers. To find the total parts of a ratio, ____ add all the parts together.  Ratios 28) In a pet shop, the ratio of fish to hamsters is 25:95. What is the simplified ratio? HCF of 25 and 95 is 5 25:95 = 5:19 29) Which two numbers have a sum of 500 and their ratio is 1:4? 1+4=5 500/5 = 100 100 x 1 = 100 100, 400 100 x 4 = 400  Ratios 30) Paul is 100 years old and Sam is 75 years old. The ratio of Paul to Sam’s age is: HCF of 100 and 75 is 25 100:75 = 4:3 31) John and Mary have 500 candies. They divide them into the ratio 3:2, respectively. How many candies does Mary have? 3+2=5 500/5 = 100 100 x 2 = 200 candies  Ratios 32) A roller coaster can seat 100 people and the ride takes 15 minutes. i. How many minutes are in 3 hours? ii. How many people can ride the roller coaster in 3 hours? 100 people in 15 minutes 3 x 60 = 180 minutes 1,200 people in 180 minutes Percentage Changes  Percentage Change divide by 100 to get 1%, then To find a percentage, ______ multiply by the percentage you want to find. For percentage change new - original x 100 use the formula: original Use the formula to find the percentage change: $50 to $90 12 to 9 90 - 50 9 - 12 x 100 = 80% x 100 = -25% 50 12  Percentage Change 33) The price of a toy decreased from $220 to $80. What is the percentage change? 80 - 220 x 100 = -63% 220 34) The price of a game increased from $120 to $150. What is the percentage change? 150 - 120 x 100 = 25% 120 Conversions  Units and Conversions Units are used to measure things like length, weight, area and volume. Metric units are all multiples of 10. decimal point To convert metric units, move the _______ and change the units. Left To go UP a unit, move the decimal to the ____. To go DOWN a unit, move the decimal to the Right ____. Convert the units: 310cm 3.10m to cm 5000ml 5l to ml 0.3kg 300g to kg Powers and Roots  Powers Index notation (powers) is a way to show repeated multiplication of numbers. multiplying a number by itself many times, When __________ simplify with powers (small number at the top right). square and #3 is called a _____. #2 is called a _______ cube Write the power and calculate the answer: 9 9=x 981 2 5 x =5 x125 53 5 22x42= x216 x2  Roots Roots are the opposite of powers. Work backwards to find what number multiplied itself equals the number. square root - a number multiplied itself 2 times. √ means ____________ cube root - a number multiplied by itself 3 times. ∛ means _________ Calculate the roots: 6 √36 11 √121 ∛28 estimate To calculate an imperfect root you will need to ________.  Powers & Roots 35) Solve 6³+7³+8³+9³+10³ 216 + 343 + 512 + 729 + 1000 = 2,800 36) Solve 3 64 =4 Extra Questions Q1. Complete the prime factor trees and identify the prime factors for A, B, C and D A. B. 16 81 2 8 3 27 2 4 2 x 2 x 2 x 2 = 16 3 x 3 x 3 x 3 = 81 3 9 2 2 3 3 C. D. 150 210 2 75 2 x 3 x 5 x 5 = 150 2 105 2 x 3 x 5 x 7 = 210 3 25 3 35 5 5 5 7 Q2. A flight from Ho Chi Minh City to Hanoi takes 200 passengers and it takes an hour and a half to reach its destination. i. How many minutes are in an hour and a half? 1.5hours x 60 minutes = 90 minutes ii. If there are flights from HCMC to Hanoi once every hour, how many passengers can travel in 3 hours? 3 flights x 200 passengers = 600 passengers iii. If a flight is at 80% capacity (20% empty seats), how many passengers will be on the flight? percentage to decimal = 80% to 0.8 0.8 x 200 passengers = 160 passengers iv. If a flight is experiencing bad weather and the flight takes 50% longer than usual, what time will Flight VN123 arrive in Hanoi if it leaves Ho Chi Minh City at 13: 20 ?? 100% + 50% = 150%, percentage to demical = 1.5 1.5 x 1.5 hours = 2.25hrs = 2 hours 15 minutes 13:20 + 2:15 = 15:35 arrival time v. If the average cost of a plane ticket is $24 per person, how much money does the airline receive for a full flight? $24 x 200 passengers = $4,800 vi. Flight #1 is at 90% capacity; flight #2 is at 75% capacity; flight #3 is at 60% capacity. How much money does the airline receive from these 3 flights at an average plane ticket of $24 per passenger. #1 90% of $4,800 = 0.9 x 4800 = $4,320 #2 75% of $4,800 = 0.75 x 4800 = $3,600 #1 60% of $4,800 = 0.6 x 4800 = $2,880. TOTAL = #1 + #2 + #3 = 4320 + 3600 + 2880 = $10,800 vii. The average capacity on the plane is 143 passengers on the weekdays and 188 passengers on the weekends. What is the percentage change of passengers from the weekdays to the weekend? (Give your answer to one decimal place) (188 - 143) X 100 = 31.5% 143 End of Class