BASIC MATHEMATICS. WORKBOOK Volume 2

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1 BASIC MATHEMATICS WORKBOOK Volume Veronique Lankar A r ef resher o n t he i mp o rt a nt s ki l l s y o u l l ne e d b efo r e y o u ca n s t a rt Alg e b ra. This can be use d a s a s elf-teaching guide o r a s s t u d y g u i de. Use i t w it h a t e x t book ( o r i n structo r!) fo r mo r e p r a ct i ce. Use t he b la nk p a g e s fo r s c ra p s. Dr. VERONIQUE LANKAR haplosciences@gmail.com 1

2 Contents Decimals Meaning of decimals, Place value 3 Meaning of the decimal system, Pizzas! 4 Place value, Adding decimals, Subtracting decimals 5 Writing decimals, Fractions to decimals, Decimals to fractions 6 Dividing, Multiplying by powers of ten, Multiplying Decimals 7 Rounding decimals, comparing Decimals 8 Dividing Decimals 9 Meaning of percents, Dividing decimals 10 Percent, word problems 12 Application:Geometry 13 Ratios, rates, proportions, percents Ratios and rates 14 Solving proportions 15 Discounts 16 Percent word problems 17 Review 20 2

3 MEANING OF DECIMALS PLACE VALUES 2006 Vero nique La n ka r Veron iq u e Lankar, Here is a centimeter ruler. Each centimeter is divided into 10 equal parts (or 10 mm). Remember mixed numbers? The Length of this strip is: two centimeters and six tenths of one centimeter (6 x 1/10). Or 2 cm and 6/10 cm or 2 6/10 cm or 2cm + 6/10 cm. It has the same meaning. Let's say the strip is really between 2 6/10 and 2 7/10. We could divide each mm into 10 more equal parts To get a better precision. We have now 100 subdivisions of each cm. Our length can be written as : 2 cm and 65/100 cm or two centimeters and sixty-five hundredths or 2 65/100 or /100 Or we could write: Length = 2 + 6/10 + 5/100. Same thing. Instead of writing 2 + 6/10 + 5/100 we use a new code: This is a decimal. It has a whole part (2) and a decimal part (65). The decimal part is made of tenths, hundredth, thousandth, ten thousandth The decimal point. Separates the whole part on the left from the decimal digits on the right. It means "and". Try to make sense out of the decimal: Find its expanded from. A table can help you: 1000 thousands 100 hundreds 10 tens 1 units Tenths 1/10 Hundredths 1/ Thousandths 1/ has 5 hundreds, 4 tens, 3 units and 4 tenths 5 hundredths 2 thousandths Or = 5 x x /10 + 4/ /1,000 We read: five hundred forty-three and four hundred fifty-two thousandth Our number system is decimal because numbers are written as powers of ten and tenths. Write word names for decimals Example nine hundred eighty-seven and ninety-three hundredth : Three tenths : 0.3 ten tenths : 1.0 Five tenths Four hundred and one hundred seventy-five thousandths Five and fifty-five thousandths Ten and four hundredths Nine and two hundred thirty-eight thousandths 3

4 MEANING OF THE DECIMAL SYSTEM PIZZA! For those who still have a hard time with the decimal part, here is some insights using pizza. 1 Let s say you like square pizzas! In that world of square pizzas 342 means: 3 packs of 100 pizzas and 4 packs of 10 pizzas and 2 left over! or Or 342 = 3 x x Imagine you don t want to eat the whole pizza anymore. Only parts of the whole pizza. Since we use the decimal system, divide the pizza into 10 slices. 0.1 Of course, each part of the pizza is now smaller than the whole pizza. Each part = one tenth of the pizza or 1/10 of the pizza (whole) Or just 1/10. In our decimal system we write 1/10 = is a decimal. It means one tenth. 1 is a decimal digit. Use your calculator: 1 10 = If you eat 3 parts of the pizza, you eat three tenths of the pizza or 3/10 In our decimal system, we write: 0.3 Use your calculator to divide 3 by ten. What do you get? 3 10 = 0.3! The parts of the pizza left can be written as: 7 tenths or 7/10 or 0.7 Use your calculator to add : What do you get? Consider these pieces of pizza. Follow the example to describe your dinner 6 different ways. Example: This is 2 and 3/10 or 2 3/10 Or 2.3 or 23/10 or Two and three tenths or twenty-three tenths 4

5 MEANING OF THE DECIMAL SYSTEM PIZZA! You may want to eat even smaller pieces. You may want to cut each slice in 10 more pieces. If you do that, each pizza will have 100 pieces. Each piece is: 1/100 of the pizza or one hundredth or 1/100. 1/100 is also written as decimal : 0.01 Check with you calculator: = /10 or 0.08 or 80/ tenths of the pizza ( 8 slices) or 80 hundredths (80 pieces) or 8/10 or 80/100. This amount can be written as a decimal : 0.08 with you calculator: = The pieces left over = 2 tenths or 2/10 20 hundredths or 20/ Consider these pieces of pizza. Follow the example to describe your dinner 6 different ways. Example: One and two tenths and three hundredths or 1 + 2/10 + 3/100 or one and twenty-three hundredths /100 or 1 23/100 or 1.23 Consider = check with you calculator if this is true. Now break the same way: = We can also write in expanded form: = 3 + 2/10 + 4/ /1,000 Write in expanded form: =

6 PLACE VALUES ADDING DECIMALS SUBSTRACTING DECIMALS Write the word name for the number Example : two and three hundred thirty-eight thousandths Write decimals in expanded form Example = / / / 1000 = / / Adding decimals is like adding whole numbers: Add hundredths, tenths, units, tens from right to left. Carry if necessary. For example: = 0.9. Because 4 tenths + 5 tenths = 9 tenths = 1.3 because 12 tenths is one unit and 2 tenths. Remember mixed numbers? 12/10 = 1 2/10 Add the decimals Example = 7.12 go from right to left: = 12 carry the = 11 carry the = When you are adding decimals with different number of digits in the decimal part, attach zeros to hold the places: = is replaced by = Subtracting decimals is like subtracting whole numbers. From right to left, you subtract thousandths, hundredths You borrow to the digit to the left if necessary. Attach zeros so both numbers have the same number of decimal digit = is replaced by = Review: 3 2/ /4 1/2 + 2/3 + 1/4 6

7 DECIMALS TO FRACTIONS FRACTIONS TO DECIMALS Write the decimal as a mixed number. Write the word name for the decimal. Example 5.42 = 5 42/100 or five and forty-two hundredths (or /100) (or 2 + ) ( or ) ( or ) Write the decimals as a mixed number. Reduce the fraction part if possible Example 0.5 = 5/10 = 1/2 (divide by 5 the top and bottom) 3.45 = 3 45/100 = 3 9/ Write the decimals as a fraction. Reduce if possible. Check with your calculator. Example 0.45 = 45/100 = 9/20 and 9 20 = 0.45 check! 4.45 = 4 45/100 = 445/100 (mixed to improper) = 89/20.and = 4.45 check! = 45 / 10,000 = Write each fraction as a decimal. Use mixed number if it helps. Write the decimal in words. Example 6/100 = 0.06 or six hundredths 509/100 = 5 9/100 (from improper to mixed) = 5.09 or Five and nine hundredths 6/10 104/10 509/10 75/10 52/ /10 23/10 7/1000 Write each fraction as a decimal: First find the equivalent fraction that has a denominator of 10,100, Example 1/5 = 2/10 multiply the numerator and the denominator by 2. 1/5 = 2/10 = 0.2 (or two tenths). 3/20 7/50 5/8 4/5 4/200 7/500 Write each mixed number as a decimal. First finds an equivalent fraction for the fraction part with 10, 100, 1000 as denominator. Example 2 2/20 = 2 10/100 = /50 4 1/8 5 1/20 5 2/5 1 2/5 4 1/200 7

8 DIVIDING, MULTIPLYING BY POWERS OF 10 MULTIPLYING DECIMALS To divide a decimal by a power of ten, move the decimal point to the left as many places as there are zeros in the divisor. Add zeroes to hold the place. 89/10 = 8.9 but 89/1000 = The zero left to 8 holds the tenth place. 0.89/10 (or ) = too. When you multiply, you move the decimal point to the right. Use zeros to hold place values x 10 = 8.9 but 89 x 1,000 = 89,000. The decimal point is between the place for the units and the place for the tenths x 1000 = 890. Multiply or divide. Write zeroes as needed. Example 100 x 0.7 = 70 and 7/100 = x 10 85/ x x 1,000 62/1, x / 1, /1,000 2/ x x x x 10 To multiply decimals (without a calculator!). Ignore the decimal point. Multiply the digits. The total of decimal digits in the answer = total decimal places in the numbers being multiplied x 0.02 : first 21 x 2 = 42 (ignore the decimal point) 0.21 has 2 decimal digits, 0.02 has 2 decimal digits. The product has therefore have 4 decimal digits x 0.02 = Multiply the decimal. Example 23 x 0.01 =.23 ( 2 decimal digits) 21.4 x x x x x x x x Did you notice? 354 x 0.1 = 35.4 and 354/10 = 35.4 (move the decimal point 1 place to the left) We have: 354/10 = 354 x 0.1. This is because 0.1 = 1/10 check with your calculator 1 10 = 0.1 So 354 x 0.1 = 354 x 1/10 = 354/10 = 35.4 or three hundred fifty-four tenths Multiplying by 0.1 is like dividing by 10. Example 3.54 x 0.1 = Multiplying by 0.01 is like 567 x 0.01 Multiplying by is like 342 x Review: Factors of 32 Prime factorization of 32 Divide 18/7 12/5 Multiply 18/7 x 12/5 Subtract 18/7-12/5 Convert the mixed numbers into a improper fraction: 3 4/10 5 4/100 Round 245,763,132 to the nearest million How many hours is a week 8

9 ROUNDING DECIMALS COMPARING DECIMALS Remember the place values?.. thousands, hundreds, tens, units. tenths, hundredths, thousandth.. To approximate a decimal you can cut his decimal part and keep only a given number of digits after the decimal point. For example: round to the nearest tenth? (it means you only want to keep one decimal digit) First find the rounding digit. Here we are looking for the tenths digit. It is 9. (65.974). Look at the next digit to the right. Here it is 7. Is this number is 5 or more? If yes add 1 to the tenths digit and omit the digits to the right to this rounding digit. We have now: 66.0 (9 +1 = 10). If the digit to the right is smaller than 5, keep the rounding digit the same becomes If the decimal to round were the approximation is 65.6 Round to the nearest tenth Example = = 80.0 keep a zero to hold the place! Round to the nearest hundredth Example = 8.34 and = Round to the nearest whole number Example 8.8 = 9. and = Round to the nearest thousandth To compare decimals: Like whole numbers. Compare from left to right (compare the hundreds, tens, units, tenths, hundredths. ). if necessary, attach zeros so decimals have the same number of decimal digits. Compare the decimals by inserting < or >. Example < because 9 > 7 (thousandths place) and and and 3.5 Word problems: About.12 of each ounce of fruit juice is sugar. How much sugar would someone drink in a 5 ounces of juice? About.40 of each ounce of orange juice is pulp. How much pulp is in a fruit that includes 1..5 ounces of orange juice? Who much sugar in 2 12-ounce cans of soft drink? 9

10 DIVIDING DECIMALS Let s say you want to evaluate a fraction with a numerator smaller than the numerator. Like 3/4 or 3 4 (you know it s 0.75 right?). The answer has to be smaller than 1. In the decimal system we are using, the quotient will be written as: 0.decimal part. We need to find the decimal part (75 in our example). It s very easy. attach additional zeros (s) at the end of the dividend (3) to make the division possible. If you have a remainder keep adding zero to the dividend (or remainder) to continue long division. Stop when there is no remainder or when you have reached the right approximation (by rounding the quotient). Let s find the decimal digits of our example 3/4. Attach zeros zero to 3 (becomes 30) and let s divide by = 7R2 You have a remainder so attach another zero (to the remainder or to the dividend) = 25 R0 (because 20 4 = 5). There is no remainder. So 3/4 = 0.75 Let s take another example. You buy 200 Yugioh cards for $8. How much cost each card? You want to keep only 2 decimal digits (stop at the nearest cent or nearest hundredth). 8/200 is smaller than one. The quotient is 0. We need to find the decimal digits. Attach a zero to 8 80/200 = 0 with remainder (80). The answer is now 8/200 = 0.0 Attach another zero to /200 = 4 with no remainder (R0). The answer is no 8/200 = 0.04 with 2 decimal digits 8/200 = cents per card! Is it a good deal? Divide. Round to the nearest hundredth Example 1/10 = 0.1 because 10/10 =1 4/5 1/3 2/3 1/5 1/4 1/80 3/5 6/12 1/10 6/8 2/500 15/60 If the numerator is larger than the denominator you can use mixed numbers to divide: 5/4 = 1 1/4 or ( 1 + 1/4) since 1/4 = 0.25 we have 5/4 = 1.25! That simple! Divide. Round to the nearest hundredths. Example: 322/22 = 14 14/22 or ( /22) = or 0.64 (to the nearest hundredths) So 322/22 = We have a whole part and a decimal part (the fraction part in the mixed number) 55/22 22/11 11/3 49/20 7/4 7/5 34/10 132/25 Review: 2 4/9 + 3/5-2 2/3 (5-2/5) 2 - (2/5) 10

11 DIVIDING DECIMALS MEANING OF PERCENTS If you divide decimals with non zero decimal digits. Like (or 3.4 / 5). Here is the trick. Multiply the numerator and denominator by a power of ten (10 or 100 or 1,000 ) so you get an equivalent fraction with whole Parts in the dividend and divisor. 3.4 x 10 = 34 and 5 x 10 = 50. So 3.4 / 5 =34 / 50 Then just follow the same procedures as before. 34 / 50 = The quotient is smaller than 1 because the numerator is smaller than the denominator. If you have replace 34.5 / 5 by 345 / 50 (multiply by 10) 345/50 = 6 45/50 then divide 45 by 50 by attaching zeros to the dividend. 45/50 = 0.9 So 345/50 = 6.9 If you have decimals at the numerator and denominator, multiply by a power of ten to get rid of the decimal parts = 4.55 / 45.2 = 4550 / Find the equivalent fraction so you don t have decimal parts any more Example = 4.5 / 3.4 = 45/ / / / 46 5 / / /14 2 /0.1 5 / / 22 The symbol % is read as percent. Percent means per hundred or out of one hundred. Therefore, 25% means 25 per hundred, 25 out of 100, or 25 hundredths or 25/100 or It is a ratio. To change a percent to a decimal, move the decimal point 2 places to the left, fill with zeros to hold the places and drop the percent sign %. 25% = 25/100 = % = % =3 0.55% = ,444% = Change percent to a decimal and then to a fraction. Simplify. Example 52% = 0.52 = 52/100 = 13/25 1% 95% 44% 109% 12% 137% Change each decimal to a percent Example 0.25 = 25/100 = 25% Change each fraction to a percent Example 2/5 = 40/100 = 40% 3/4 = 0.75 = 75/100 = 75% 1/4 = 5/12 7/20 3/8 3/5 5/16 Review Find the LCM of 15, 12, 6 Evaluate 1/15 + 5/12 + 1/6) 11

12 MEANING OF PERCENTS WORD PROBLEMS Fill the blanks. Use a calculator. Fraction Decimal Percentage Fraction Decimal Percentage 1/2 (or 1 2) 1/6 1/3 ( or ) 1/7 1/4 (or ) 1/8 1/5 1/9 1/10 1/12 1/15 1/20 If socks cost $8.97 for 3 pairs, how much does one pair cost? If candy bars are 6 for $2.58, how much is one candy bar? Jimmy had to divide 2.5 candy bars with Peter. How much of a candy bar would they each get? For an art project, 5 boys divided 530 beads. How many beads did they each get? The body can burn only of an ounce of alcohol an hour. If an average-sized person has one drink, his or her blood alcohol concentration (BAC) is how many hours will it take his or her body to clear that much of alcohol from the blood? If someone has 3 drinks in 1 hour, the BAC rises to How many hours it will take to burn 3 drinks? A computer printer take of a second to print a letter. How long would it take to print the word Technology? Find One tenth less than 0.7 Write sixty-five hundredths Write twenty-nine and seventy-four hundredths Write thirty-six and seventy-four hundredths Blue bridge is.45 miles long, while Yellow Bridge is 1.23 miles long. How much longer is Yellow Bridge? 12

13 REVIEW GEOMETRY Let s review order of operations ( parentheses, exponents, divide or multiply, subtract or add) Evaluate x (9-10) ( ) ( ) you can replace by a fraction bar 4/5. (2/3) - 1/3. (3/5) Convert 4 1/6 to a decimal. Round to the nearest hundredth. Compare 5/6 and 0.32 Compare 0.63 and 3/8 From Volume 1, you know: area inside a figure = number of square units inside a figure. Area of a rectangle = width x length. (example: the area of a square 3 in. wide = 3x3 = 9 in 2 ) What is the area of a circle that has a given diameter. (diameter = radius x 2 see figure - left)? diameter radius The exact formula is area = pi x (radius) 2 You could fit the circle in square to find an estimate of the Area. Suppose we have a circle that has a radius of 3 units. In that case, the area of the square is: 6 x 6 = 36 units 2 The side of the square = 2 x 3 = 6 units (the diameter). So: estimate (36 units 2 ) = 2 x 3 x 2 x3 = 4 x 3 2 = 4 x (radius) 2 The exact value is less (the corners of the square are to be cut) A better estimate is: area = 3.14 x (radius) 2 Or area = (22/7) x (radius) 2 pi is called an irrational number. Because it has an infinite number of decimal digit. Pi = units Estimate area = 36 squares Find the area of a circle that has a radius of 3.5 feet? A circular canvas net used by firefighters has a radius of 2.1 meter. What is the area of the net? In Volume 1 the perimeter was the distance around a polygon. The distance around a circle is called the circumference. With a string, find the circumference of a soda can. Then estimate the diameter of the can with the string. You should find that the circumference is about = diameter x 3.14 C = pi x diameter is the exact formula. Example: our previous circle has a circumference of 6 x 3.14 = units or 19 units to the nearest one digit. The wheel of a bicycle has a radius of 35 cm. What is the circumference of the wheel? What is the circumference of a circle that has a diameter of 7 inches? 13

14 RATIOS... RATIOS UNIT RATES PROPORTIONS A ratio is a comparison of two quantities by division. A ratio can be written is 3 different ways. Suppose there are 15 boys for 3 girls in our classroom. This can be written as the ratio: 15/3 or 3/1 or 3 to 1 or 3:1. The ratio can be read as 3 to 1. Consider the figure: The ratio of shaded area to total area is 4:12 or 1:3 (with 4/12 = 1/3 ) The ratio of unshaded are to total area is The ratio of shaded area to unshaded area is Write each ratio as a fraction. Simplify the fraction if possible. (that is the fraction is in simplest form) Example 45 to 1 (or 45:1) can be written as 45/1 3 to 15 5 to 10 9:27 Express each ratio in simplest form Example 8 out of 10 people ratio = 8:10 or 8/10 or 4/5 (divide by 2 the numerator and denominator) 8 is to 10 what 4 is to 5. You can use your calculator and check: 8/10 = 0.8 and 4/5 = out of tickets:60 tickets $5 out of $100 2 hours: 20 hours A rate is a ratio that compares two different units: $20 for 2 hours of work The rate is $20 / 2h or $10 / 1 h or $10 per hour or $10 each hour Express the rate in simplest form Example $44 for 10 hours the rate is $44 / 10h = $22 / 5 hours 81 men for 10 women 45 cats for 10 dogs 80 miles for 3 hours $100 for 30 hours A unit rate is a comparison to 1 unit. For example 60 miles per hour is a unit rate because it compares 60 miles to 1 hour. The unit rate is written as: 60 miles/hour Express each rate as a unit rate Example: $45 for 9 hours $45/9hours = $5/hour just divide 45 by 9 $330 for three days 250 kilometers in 5 hours 88 pounds in 11 weeks 483 miles in 12 hours $15 for 35 newspapers 300 miles in 6 hours A proportion is an equation that shows that two ratios are equivalent. To determine if a pair of ratios form a proportion, cross multiply to find the cross products. Example 1/5 = 20 /100 because 100 x 1 = 5 x 20 Determine whether each pair of ratios forms a proportion. 12/30 and 28/70 16/6 and 10/3 10/24 and 6/14 Is 5/8 =10/18 a true proportion? Is 66miles/4gallons = 33 miles/ 2gallons a true proportion? 14

15 RATIOS... PERCENTS SOLVING PROPORTIONS A percent is a ratio that compares a number to 100. For example 1 is to 4 (1:4) what 25 is to 100 (25:100). The fractions 1/4 and 25/100 are equivalent fractions. 1/4 = 25/100 = Remember? 0.25 = 25/100 = 25% or 25 percents or 25 hundredths. 1/2 and 50/100 are also equivalent fractions. 1/2 = 50/200 = 0.5 = 50% or 50 percents or 50 hundredths. Express each ratio or fraction as a percent. You can use your calculator Example: 16 out of 25 divide to find the decimal 16/25 = 0.64 = 64/100 = 64% 2 in 50 9 in out of out of in 12 1/5 Review: Express each percent as a fraction in simplest form Example: 25% = 25/100 = 1/4 85% 3% 10% 56% Sometimes one of the number in a proportion is unknown. In this case, it is necessary to solve the proportion. To solve a proportion, find a number to replace the unknown so that the proportion is true: cross multiply. Example: Solve: 9/6 = 3/n. Find the cross products: 9n = 18. Divide 18 by 9 to find n. n=18/9 = 2. Check = 9 x 2 = 3 x 6 = 18 Solve each proportion. Write the ratio as a percent. Use you calculator to divide. Example: 1/4 = m/8 4m = 8 so m=2 check: 4 x 2 = 8 x 1 = 8 Then 1 4 = 2 8 = 0.25 and 0.25 = 25% 2/24 = t/48 2/24 = % t/3 = 10/15 10/15 = % m/4 = 7/14 n/12 = 25/60 In a percent problem, you can find 3 kinds of questions: - Find a percent of a whole. Example: What is 25% of 4? The answer is 1. - Find what percent one number (the part) is of another number (the whole). 1 is what percent of 4? 25% - Find the whole when the percent of it is given as a part. 1 is 25 % of what number? 4 Here is a formula you can remember to solve such problems: : parts = is = % whole of 100 is (parts) must be replaced by the number that follows (or precede) the word is in the question. of (whole) number that follows the word of in the question. It is also the whole. % the number in front of the % or word percent in the question. Solve the proportion to find the unknown. 15

16 RATIOS... PERCENTS DISCOUNTS Find the percent of the number Example: 1% of 50. Using the formula: parts/50 = 1/100 solve for parts. parts = 50/100 = % of % of /2 % of 100 change first into a decimal 0.5% of 40 What percent is one number of another? Example: 10 is what % of 40? 10/40 = % / 100 solve for %. % = (10 x 100) / 40 = 25 % 6 is what % of 12? 66 is what percent of 11? 3.5 is what percent of 100? 50 is what percent of 50? Find the whole when a percent is given Example: 100% of what number is 3? 3 / whole = 100/100 = 1 so whole/3 =1 or whole = 3! 50% of what number is 45? 500% of what number is 5? 10% of what number is 10? 20% of what number is 200? A real estate agent has a commission rate of 6%. If she sells a house for $124,000: What is her commission? A tire salesman has a 12% commission rate. If he sells a set of radial tires for $400: What is his commission? Review: 12 is what percent of 60? What percent of 64 is 288? 98 is what percent of 140? What is 90% of 10? The amount by which a price is reduced is called a discount. To find the amount of a discount, multiply the regular price by the discount rate. To find the sale price, subtract the discount from the regular price. For example: Regular price = $27 Discount = 20% Amount of discount = (20x27) /100 = $5.40 So Sale price = $27 - $5.40 = $21.60 you could have take a short cut: $27 x 80% with 80 = Find the amount of discount, the sale price and check using the short cut $25 and 10% discount Discount Sale price short cut : Sale = (25 x 90) 100 $220 and 60% discount Discount Sale price short cut : Sale Solve the proportion: 1:15 and t:30 16

17 RATIOS... PROBLEMS $160 and 12 1/2 % discount (with 12 1/2 = 12.5 ) Discount Sale price short cut : Sale Find the discount rate for this item: (in %) $50 shoes for $40 PROBLEMS: A pole is 100 feet long. 25% of the pole is under the ground. What is the length above the ground? An organization spent $2940 for administrative expenses. This amount is 12% of the money it collected. What is the total amount of money the organization collected The Fire department received 24 false alarms out of a total of 200 alarm received. What percent of the alarms received were false alarms? To find the final price of an item you need to add sales tax. To find the tax amount, multiply the price before tax by the tax rate. Add the tax amount to the price to find the final cost. For example: Price = $125 and Tax = 6% Amount of tax = $ (125 x 6) /100 = $7.50 So Price TTC = $125 + $7.50 = $132.5 short cut: Price TTC = $125 x 106% with 106 = Find the amount of tax, the price TTC and check with the short cut $250 and 8% tax Amount tax Sale price short cut: TTC = (250 x 108) 100 $800 and 4 % tax Amount tax Sale price short cut: TTC $10 and % tax with 6.125% = To find discounts or taxes you could also use the formula: parts = % whole 100 5% of 200 is? Solve for parts. parts/200 = 5/100 so parts = 5/100 x 200 (or 5% x 200) = 10 The price of a dinner is $200 before tax and $210 TTC. What is the tax in %? 8 out 25 employees work part-time. What percent of the employees work part-time? You saved 15% of your salary. What fraction of your salary you save? Write 15% as a fraction. Reduce the fraction (divide the numerator and denominator by the same number). 17

18 RATIOS... PERCENTS WORD PROBLEMS You solved 5 problems out of 8. Write this fraction as a decimal. You borrowed $12,000 at an interest rate of 10% per year. interests are computed each year This means that, after 1 year, you will owe $12, % of $12,000 or 12, ,000 x 10 /100 After 1 year you owe After 6 months you owe After 3 years you owe after 3 months you owe A movie theater increased the price of admission by 20%. Tickets had sold for $6.50. What is the current price? Review: Find the LCD and add the fractions: 2/45-1/9 + 3/5 3/32 15/8 3 (2) Change the % into a fraction: 300% Write 1/10,000 using negative exponent Write 435,000,000 in scientific notation Divide 453 by 1,000 Multiply 453 by 1,000 Round 435 to the nearest ten Round to the nearest tenth Reduce the following fractions to the lowest terms: 2/2 3/9 8/12 5/10 10/ /10 4/16 25/75 Change the following mixed numbers to improper fractions: 2 1/3 11 1/9 25 2/3 5 1/2 4 1/2 3 3/4 16 1/6 7 1/4 Find the value of t in the following proportions: 5/15 = 6/t t/100 = 500/20 Change to fractions (reduce to lowest terms): Change to decimal form: 1/5 1/4 1/50 1/1,000 Change the percentage form to its decimal form: 25% 10% 50% 20 % 18

19 MORE PERCENT PROBLEMS On new year s Eve, you made a resolution to lose 30 pounds by the end of March. And sure enough, your weight dropped from 140 to 110. By what percentage did you weight fall? Hint: percentage of change = change / original number If Melanie Shapiro was earning $100 and her salary were cut to $93, by what percent was her salary cut? Hint: percentage of change = change / original number A sofa was marked with the following sign: The price of this sofa has been reduced by 23%. You can save $138 if you buy now. What was the original price of the sofa? Hint: discount = ( selling price ) x ( discount rate ) with discount rate = 23/100 = 0.23 and discount = $128 John earns a commission of 38% of the cost of every et of encyclopedia that he sells. Last year he earned $4560 in commissions. What was the cost of the encyclopedias that he sold last year? Hint: commission = (cost) x ( commission rate) Hector received a pay raise this year. The raise was 6 % of last year s salary. This year he will earn $15,900. What was his salary last year before the raise? Hint: (new salary) = (old salary) + (old salary) x (rate) = (old salary) ( 1 + r ) Robert invested some money at 8% simple interest. At the end of the year, the total amount of his original principal and the interest was $7560. How much did he originally invest? Hint: Total = (original) + (original) x (interest rate) = ( original) x ( 1 + interest rate ) Find the interest on $7,000 borrowed at a simple interest rate of 12% for one year. Hint: interest = (investment) ( rate) (time in year) 19

20 REVIEW REVIEW Add the following fractions and reduce it to lowest terms: 2 1/5 + 1/5 2 1/ /3 1/2 + 1/ /3 4 2/ / /12 Subtract the following fractions and reduce to lowest terms: find the LCD 3/4-1/4 1/8-1/ / /3 20 1/2-8 1/6 1/8-1/12 2 1/8-1/12 Multiply these fractions and reduce the answers to lowest terms: 2/4 x 1/2 5 1/6 x 1/8 1/8 of 1/12 1/2 x 1/3 x 1/4 1/2 of 1/4 2 2/3 x 4 1/2 Divide these fractions and reduce to the lowest terms: 1/4 1/2 1/20 1/3 10 1/2 1/3 1/150 1/2 Evaluate: , x x x Prime factorization of 60 LCM of 5, 15, 10 20

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