Decimals Both decimals and fractions are used to write numbers that are between whole numbers. Decimals use the same base-ten place-value system as whole numbers. You can compute with decimals in the same way that you compute with whole numbers. Some fractions between and 2: 3 2, 7 4, 8 Some decimals between and 2:.5,.75,.375,.,.999, 7, 6 3 6 32 Decimals are another way to write fractions. Many fractions have denominators of,,,, and so on. It is easy to write the decimal names for fractions like these. Both decimals and fractions are used to name part of a whole thing or part of a collection. Decimals and fractions are also used to make more precise measurements than can be made using only whole numbers. Fractional parts of a dollar are almost always written as decimals. This square is divided into equal parts. Each part is of the square. The decimal name for is.. 4 of the square is shaded. The decimal name for 4 is.4. This square is divided into equal parts. Each part is of the square. The decimal name for is.. 42 of the square is shaded. 42 The decimal name for is.42. Like mixed numbers, decimals can be used to name numbers greater than one. 26 twenty-six
In a decimal, the dot is called the decimal point. It separates the whole-number part from the decimal part. A decimal with one digit after the decimal point names tenths. A decimal with two digits after the decimal point names hundredths. A decimal with three digits after the decimal point names thousandths. tenths hundredths thousandths.4 4.34 3 4.62 6, 8 75 3.8.75.3,.9 9.3 3.98 9, Reading Decimals One way to read a decimal is to say it as you would a fraction or mixed number. For example,., and can be read 9 as one-thousandth. 7.9 7, so 7.9 can be read as seven and nine-tenths. You can also read decimals by first saying the whole number part, then saying point, and finally saying the digits in the decimal part. For example, 6.8 can be read as six point eight ;.5 can be read as zero point one five. This way of reading decimals is often useful when there are many digits in the decimal. 2 8 Decimals were invented by the Dutch scientist Simon Stevin in 585. But there is no single, worldwide form for writing decimals. For 3.25 (American notation), the British write 3. 25, and the Germans and French write 3,25..8 is read as 8 hundredths or zero point one eight. 24.5 is read as 24 and 5 tenths or twenty-four point five..8 is read as 8 thousandths or zero point zero zero eight. Write a decimal for each picture.. 2. Read each decimal to yourself. Write each decimal as a fraction or mixed number. 3..7 4. 4.56 5. 24.68 6..4 Check your answers on page 434. twenty-seven 27
Extending Place Value to Decimals The first systems for writing numbers were primitive. Ancient Egyptians used a stroke to record the number, a picture of an oxbow for, a coil of rope for, a lotus plant for,, and a picture of a god supporting the sky for,,. This is how an ancient Egyptian would write the number 54: Our system for writing numbers was invented in India and later improved in Arabia. It is called a base-ten system. It uses only symbols, which are called digits:,, 2, 3, 4, 5, 6, 7, 8, and 9. In this system, you can write any number using only these digits. It should come as no surprise that our number system uses exactly symbols. People probably counted on their fingers when they first started using numbers. For a number written in the base-ten system, each digit has a value that depends on its place in the number. That is why it is called a place-value system. In the number 7,86, 7 is in the thousands place; its value is 7 thousands, or 7,. is in the hundreds place; its value is. 8 is in the tens place; its value is 8 tens, or 8. 6 is in the ones place; its value is 6 ones, or 6. The in 7,86 serves a very important purpose: It holds the hundreds place so that the 7 can be in the thousands place. When used in this way, is called a placeholder. 28 twenty-eight
The base-ten system works the same way for decimals as it does for whole numbers. In the number 47.85, 8 8 is in the tenths place; its value is 8 tenths, or, or.8. is in the hundredths place; its value is. 5 5 is in the thousandths place; its value is 5 thousandths, or,, or.5. In the number 4.36, 3 3 is in the tenths place; its value is 3 tenths, or, or.3. 6 6 is in the hundredths place; its value is 6 hundredths, or, or.6. is in the thousandths place; its value is. Right to Left in the Place-Value Chart Study the place-value chart below. Look at the numbers that name the places. As you move from right to left along the chart, each number is times as large as the number to its right. one ten, s one ten s one ten s one ten s one ten s one, ten s twenty-nine 29
You use facts about the place-value chart each time you make trades using base- blocks. Suppose that a flat is worth. Then a long is worth, or.; and a cube is worth, or.. You can trade one long for ten cubes because one equals ten s. You can trade ten longs for one flat because s equals one. ten You can trade ten cubes for one long because ten s equals one. For this example: A flat is worth. A long is worth or.. A cube is worth, or.. Left to Right in the Place-Value Chart Study the place-value chart below. Look at the numbers that name the places. As you move from left to right along the chart, each number is as large as the number to its left. one of, one of one of one of one of one, of. What is the value of the digit 2 in each of these numbers? a. 2,6.8 b..2 c. 34.52 2. Using the digits 9, 3, and 5, what is a. the smallest decimal that you can write? b. the largest decimal less than that you can write? c. the decimal closest to.5 that you can write? Check your answers on page 434. 3 thirty
Powers of for Decimals Study the place-value chart. Look at the numbers across the top of the chart that name the places. A whole number that can be written using only s as factors is called a power of. A power of can be written in exponential notation. Powers of (greater than ) Standard Notation Product of s Exponential Notation * 2, * * 3, * * * 4, * * * * 5 A number written in the usual place-value way, like, is in standard notation. A number written with an exponent, like 2, is in exponential notation. Decimals that can be written using only.s as factors are also called powers of. They can be written in exponential notation with negative exponents. Powers of (less than ) Standard Notation Product of.s Exponential Notation.... *. 2.. *. *. 3.. *. *. *. 4.. *. *. *. *. 5 The number is also called a power of because. A number raised to a negative exponent is equal to the fraction over the number raised to the positive exponent. For example, 2 2 *.,s,s,s s s s.s.s.s.s.s 5 4 3 2. 2 3 4 5 All of the numbers across the top of a place-value chart are powers of. Note the pattern: Each exponent is less than the exponent in the place to its left. This is why mathematicians defined to be equal to. thirty-one 3
Comparing Decimals One way to compare decimals is to model them with base- blocks. If you don t have the blocks, you can draw shorthand pictures of them. Base- Block Name Shorthand Picture cube long flat big cube Compare 2.3 and 2.6. 2.3 2.6 2 flats and 3 longs are more than 2 flats, long, and 6 cubes. So, 2.3 is more than 2.6. 2.3 2.6 Sometimes the big cube is the whole, or ONE. Compare 2.23 and 2.74. 2.23 2.74 Each picture shows 2 big cubes. 2 flats and 3 longs are more than flat and 7 longs and 4 cubes. So, 2.23 is more than 2.74. 2.23 2.74 32 thirty-two
You can write a at the end of a decimal without changing the value of the decimal:.7.7. Writing s is sometimes called padding with s. Think of it as trading for smaller pieces..3.3.3 Trade each long.3 for cubes. Padding with s makes comparing decimals easier. Compare.3 and.6..3.3 (Trade 3 longs for 3 cubes.) 3 cubes is more than 6 cubes. 3 hundredths is more than 6 hundredths..3.6, so.3.6. Compare.97 and.. (Trade flat for cubes.) 97 cubes is less than cubes. 97 hundredths is less than hundredths..97., so.97. A decimal place-value chart can be used to compare decimals. Compare 4.825 and 4.862. s.s.s.s ones tenths hundredths thousandths 4 8 2 5 4 8 6 2 The ones digits are the same. They are both worth 4. The tenths digits are the same. They are both worth 8 tenths, or 8, or.8. The hundredths digits are not the same. The 2 is worth 2 hundredths, or.2. The 6 is worth 6 hundredths, or.6. The 6 is worth more than the 2. So, 4.862 is more than 4.825. Imagine two points that are meters apart. Sound would travel that distance in about.3 second. And a beam of light would travel that distance in about.3 second. Light travels about million times as fast as sound. Compare the numbers in each pair...59,.59 2..99,. 3. 4,.3 4..99,. Check your answers on page 434. thirty-three 33
Addition and Subtraction of Decimals There are many ways to add and subtract decimals. One way is to use base- blocks. When working with decimals, we usually use a flat as the ONE. To add with base- blocks, count out blocks for each number and put all the blocks together. Make any trades for larger blocks that you can, and then count the blocks for the sum. To subtract with base- blocks, count out blocks for the larger number. Take away blocks for the smaller number, making trades as needed. Then count the remaining blocks. Using base- blocks is a good idea, especially at first. However, drawing shorthand pictures is usually easier and faster..63 3.6? First, draw pictures for each number. Next, draw a ring around longs and trade them for a flat. This means that.63 3.6 5.23. This makes sense because.63 is near 2 and 3.6 is near 3 2. So, the answer should be near 5, which it is. After the trade, there are 5 flats, 2 longs, and 3 cubes..63 3.6 5.23 3.7 2.6? The picture for 3.7 does not show any longs. You want to take away 2.6 (2 flats and 6 longs). To do this, trade flat for longs. Now remove 2 flats and 6 longs (2.6). 3.7 2.6.47 flats, 4 longs, and 7 cubes are left. These blocks show.47. 34 thirty-four
Most paper-and-pencil strategies for adding and subtracting whole numbers also work for decimals. The main difference is that you have to line up the places correctly, either by writing s at the end of the numbers or by lining up the ones places. 4.56 7.9? Partial-Sums Method: Add the ones. Add the tenths. Add the hundredths. 4 7.5.9.6. s.s.s 4. 5 6 7. 9.. 4. 6 Add the partial sums...4.6 2. 4 6 Column-Addition Method: Add the numbers in each column. Trade 4 tenths for one and 4 tenths. Move the one into the ones column. s.s.s 4. 5 6 7. 9. 4 6 2. 4 6 4.56 7.9 2.46, using either method. 9.4 4.85? Trade-First Method: Write the problem in columns. Be sure to line up the places correctly. Since 4.85 has two decimal places, write 9.4 as 9.4. s.s.s 9. 4 4. 8 5 Look at the.s place. You cannot remove 5 hundredths from hundredths. s.s.s 3 9. 4 4. 8 5 So trade tenth for hundredths. Now look at the.s place. You cannot remove 8 tenths from 3 tenths. s.s.s 3 8 3 9. 4 4. 8 5 4. 5 5 Trade one for tenths. Now subtract in each column. 9.4 4.85 4.55 thirty-five 35
9.4 4.85? Left-to-Right Subtraction Method: Since 4.85 has two decimal places, write 9.4 as 9.4. 9.4 Subtract the ones. 4. 5.4 Subtract the tenths..8 4.6 Subtract the hundredths..5 4.55 9.4 4.85 4.55 9.4 4.85? Counting-Up Method: Since 4.85 has two decimal places, write 9.4 as 9.4. There are many ways to count up from 4.85 to 9.4. Here is one. 4.85.5 5. 4. 9..4 9.4 Add the numbers you circled and counted up by:.5 4..4 4.55 You counted up by 4.55. 9.4 4.85 4.55 Calculator: If you use a calculator, it s important to check your answer by estimating because it s easy to press a wrong key accidentally. Add or subtract. 4.62.4 2. 3.8.49 3. 2.4 2.377 Check your answers on page 434. 36 thirty-six
Multiplying by Powers of Greater than Multiplying decimals by a power of greater than is easy. One way is to use partial-products multiplication. Solve, * 45.6 by partial-products multiplication. Step : Solve the problem as if there were no decimal point. 4 * 5 * 6 * * 456 4 5 6 456 Step 2: Estimate the answer to, * 45.6 and place the decimal point where it belongs., * 45 45,, so, * 45.6 must be near 45,. Some powers of greater than : 2 * 3 * *, 4 * * *, So, the answer to, * 45.6 is 45,6. Another way to multiply a number by a power of greater than one is to move the decimal point. Think of this as a shortcut., * 45.6? Locate the decimal point in the power of.,. Move the decimal point LEFT until you get to the number. Count the number of places you moved the decimal point. 3 places Move the decimal point in the other factor the same number of places, but to the RIGHT. Insert s as needed. That s the answer. So,, * 45.6 45,6. Multiply.. * 4.56 2..28 *, 3., * $4.5 4..4 * Check your answers on page 434. thirty-seven 37
Multiplication of Decimals You can use the same procedures for multiplying decimals that you use for whole numbers. The main difference is that with decimals you have to decide where to place the decimal point in the product. One way to solve multiplication problems with decimals is to multiply as if both factors were whole numbers. Then adjust the product: Step. Make a magnitude estimate of the product. Step 2. Multiply as if the factors were whole numbers. Step 3. Use the magnitude estimate to place the decimal point in the answer. A magnitude estimate is a very rough estimate that answers questions like: Is the solution in the ones? Tens? Hundreds? Thousands? 5.2 * 3.6? Step : Make a magnitude estimate. Round 5.2 to 2 and 3.6 to 4. Since 2 * 4 8, the product will be in the tens. (In the tens means between and.) Step 2: Multiply as you would with whole numbers using the partial-products method. Work from left to right. Ignore the decimal points. 3 * 3 * 5 3 * 2 6 * 6 * 5 6 * 2 Add the partial products. 52 36 3 5 6 6 3 2 5472 Step 3: Place the decimal point correctly in the answer. Since the magnitude estimate is in the tens, the product must be in the tens. Place the decimal point between the 4 and the 7 in 5472. The price of a gallon of gasoline always includes 9 an additional cent per gallon. Many people believe that this practice is deceptive. For example, suppose that gasoline costs $2.879 per gallon. A -gallon purchase would cost exactly * $2.879 $28.79. A 9-gallon purchase should cost exactly 9 * $2.879 $25.9, but the buyer is charged $25.92. So, 5.2 * 3.6 54.72. 38 thirty-eight
3.27.8? Step : Make a magnitude estimate. Round 3.27 to 3 and.8 to. Since 3 * 3, the product will be in the ones. (In the ones means between and.) Step 2: Multiply as you would with whole numbers. Ignore the decimal points. 327 * 8 8 * 3 24 8 * 2 6 8 * 7 56 24 6 56 266 Step 3: Place the decimal point correctly in the answer. Since the magnitude estimate is in the ones, the product must be in the ones. Place the decimal point between the 2 and the 6 in 266. So, 3.27 *.8 2.66. Sometimes a magnitude estimate is on the borderline and you need to be more careful. For example, a magnitude estimate for 8.5 * 5.2 is 2 * 5. The answer may be in the s or it may be in the s. But the answer will be close to. Since 85 * 52 962, place the decimal point between the 6 and the 2: 8.5 * 5.2 96.2. There is another way to find where to place the decimal point in the product. This method is especially useful when the factors are less than and have many decimal places. 3.27 *.8? Count the decimal places to the right of the 2 decimal places in 3.27 decimal point in each factor. decimal place in.8 Add the number of decimal places. This is how many 2 3 decimal places there will be in the product. Multiply the factors as if they were whole numbers. 327 * 8 266 Start at the right of the product. Move the decimal point LEFT the necessary number of decimal places. So, 3.27 *.8 2.66. Multiply...7 * 5.7 2. 2.33 * 8.4 3..6 * 4.4 4..3 *.2 Check your answers on page 434. thirty-nine 39
Lattice Multiplication with Decimals Find 34.5 * 2.5 using lattice multiplication. Step : Make a magnitude estimate. 34.5 * 2.5 35 * 2 7 The product will be in the tens. (The symbol means is about equal to.) Step 2: Draw the lattice and write the factors, including the decimal points, at the top and right side. In the factor above the grid, the decimal point should be above a column line. In the factor on the right side of the grid, the decimal point should be to the right of a row line. Step 3: Find the products inside the lattice. Step 4: Add along the diagonals, moving from right to left. Step 5: Locate the decimal point in the answer as follows. Slide the decimal point in the factor above the grid down along the column line. Slide the decimal point in the factor on the right side of the grid across the row line. When the decimal points meet, slide the decimal point down along the diagonal line. Write a decimal point at the end of the diagonal line. Step 6: Compare the result with the estimate. The product, 7.725, is very close to the estimate of 7. 7 3 4 6 5 2 8 3 4 6 8 2 5 7 2 5 2 Steps 3 5 2 5 Steps 4 6 5 5 2 5 2 5 Find 73.4 *.5 using lattice multiplication. A good magnitude estimate is 73.4 *.5 73 * 73. (The symbol means is about equal to.) The product, 77.7, is close to the estimate of 73. 7 7 7 3 7 3 3 5 5 7 4 4 2 5 The lattice method of multiplication was used by Persian scholars as long ago as the year. It was often called the grating method. Draw a lattice for each problem and multiply.. 32.5 * 2.5 2. 4.2 * 7 3. 8. * 23.4 Check your answers on page 434. 4 forty
Dividing by Powers of Greater than Here is one method for dividing by a power of greater than. Decimals and Percents 45.6 /,? Step : Locate the decimal point in,. the power of. Step 2: Move the decimal point LEFT until you get to the number. Step 3: Count the number of places you moved the decimal point. Step 4: Move the decimal point in the other number the same number of places to the LEFT. Insert s as needed. 3 places Some powers of greater than : 2 3, 4, 5, 6,, 45.6 /,.456 35 /? 35 /,? $29.5 /,?.,.,. 2 places 4 places 3 places 35 / 3.5 35 /,.35 $29.5 /, $.29 (rounded to the nearest cent) Note: When the dividend (the number you are dividing) does not have a decimal point, you must locate the decimal point before moving it. For example, 35 35. Divide.. 56.7 / 2..47 / 3. $29 /, 4. 6 /, Check your answers on page 434. forty-one 4
Division of Decimals Here is one way to divide decimals: Step : Make a magnitude estimate of the quotient. Step 2: Divide as if the divisor and dividend were whole numbers. Step 3: Use the magnitude estimate to place the decimal point in the answer. A magnitude estimate is a rough estimate of the size of an answer. A magnitude estimate tells whether an answer is in the ones, tens, hundreds, and so on. 97.24 / 26? Step : Make a magnitude estimate. Since 26 is close to 25 and 97.24 is close to, the answer to 97.24 / 26 will be close to the answer to / 25. Since / 25 4, the answer to 97.24 / 26 should be in the ones. (In the ones means between and.) Step 2: Step 3: Divide, ignoring the decimal point. 26 9 7 2 4 78 3 924 4 4 884 78 3 4 4 4 374 9724 / 26 374 Decide where to place the decimal point. According to the magnitude estimate, the answer should be in the ones. So, 97.24 / 26 3.74. Sometimes a magnitude estimate is on the borderline and you need to be more careful. For example, a magnitude estimate for 2,89 / 3.4 is 3, / 3,. This answer is in the thousands. But the exact answer may be in the hundreds. You should place the decimal point so that the answer is close to,. Since 2,89 / 34 85, you should attach one zero, followed by a decimal point: 2,89 / 3.4 85. Divide.. 48.8 / 6 2. 25.32 / 2 3. 4.55 / 3.5 Check your answers on page 434. 42 forty-two
The answers to decimal divisions do not always come out even. When you divide as if the divisor and dividend were whole numbers, there may be a non-zero remainder. If the remainder is not zero:. Rewrite a remainder as a fraction: Make the remainder the numerator of the fraction. Make the divisor the denominator of the fraction. 2. Add this fraction to the quotient and round the sum to the nearest whole number. 3. Then use the magnitude estimate to place the decimal point in the answer. The decimal division below does not come out even. 8.27 / 4? Make a magnitude estimate. Since 8.27 is close to 8, 8.27 / 4 8 / 4. Since 8 / 4 2, the answer to 8.27 / 4 should be in the tens. (In the tens means between and.) Divide, ignoring the decimal point. The symbol means is about equal to. 4 8 2 7 8 2 27 24 6 3 26 827 / 4 26 R3. The quotient is 26, and the remainder is 3. Rewrite the remainder 3 as the fraction 3 4. Add this fraction to the quotient: 827 / 4 26 3 4. Round this answer to the nearest whole number, 27. Decide where to put the decimal point. According to the magnitude estimate, the answer should be in the tens. So, 8.27 / 4 2.7. Divide.. 8.8 / 3 2. 86.4 / 24 3. 45.2 / 3 Check your answers on page 434. forty-three 43
Column Division with Decimal Quotients Column division can be used to find quotients that have a decimal part. In the example below, think of sharing $5 equally among 4 people. 4 5?. Set the problem up. Draw a line to separate the digits in the dividend. Work left to right. Think of the in the tens column as $ bill. 4) 5 3. If 4 people share 5 $ bills, each person gets 3 $ bills. There are 3 $ bills left over. 3 4) 5 5 2 3 5. If 4 people share 3 dimes, each person gets 7 dimes. There are 2 dimes left over. Draw another line and another in the dividend to show pennies. 3. 7 4) 5. 5 3 2 28 3 2 7. If 4 people share 2 pennies, each person gets 5 pennies. 3. 7 5 4) 5. 5 3 2 2 28 2 3 2 2. The $ bill cannot be shared by 4 people. So trade it for $ bills. Think of the 5 in the ones column as 5 $ bills. That makes + 5, or 5 $ bills in all. 4) 5 5 4. Draw a line and make decimal points to show amounts less than $. Write after the decimal point in the dividend to show there are dimes. Then trade the 3 $ bills for 3 dimes. 3. 4) 5. 5 3 2 3 6. Trade the 2 dimes for 2 pennies. 3. 7 4) 5. 5 3 2 2 28 3 2 The column division shows that 5 / 4 3.75. This means that $5 shared equally among 4 people is $3.75 each. 44 forty-four
Rounding Decimals Sometimes numbers have more digits than we need to use. This is especially true of decimals. A calculator display may show seven or more digits to the right of the decimal point, even when only one or two digits are needed. Rounding is a way to get rid of unnecessary digits to the right of the decimal point. There are three basic ways to round numbers: A number may be rounded down, rounded up, or rounded to a nearest place. The examples on pages 45 and 46 involve rounding to hundredths, but rounding to any other place to the right of the decimal point is done in a similar way. See page 249 for examples that involve rounding to the left of the decimal point. Rounding Down To round a decimal down to a given place, just drop all the digits to the right of the desired place. When a bank computes the interest on a savings account, the interest is calculated to the nearest tenth of a cent. But the bank cannot pay a fraction of one cent. So the interest is rounded down, and any fraction of one cent is ignored. The bank calculates the interest earned as $7.28. Round down to the next cent (the hundredths place). First, find the place you are rounding to: $7.28. Then, drop all the digits to the right of that place: $7.2. The bank pays $7.2 in interest. Truncate means to shorten by cutting off a part. Rounding down a decimal by dropping digits to the right of the decimal point is often called truncating. Rounding Up To round a decimal up to a given place, look at all the digits to the right of the desired place. If any digit to the right of the desired place is not, then add to the digit in the place you are rounding to. (You will have to do some trading if there is a 9 in that place.) If all the digits to the right of the desired place are, then leave the digit unchanged. Finally, drop all the digits to the right of the desired place. Running events at the Olympic Games are timed with automatic electric timers. The electric timer records a time to the nearest thousandth of a second and automatically rounds up to the next hundredth of a second. The rounded time becomes the official time. minute 2.45 seconds rounds up to minute 2.5 seconds. forty-five 45
The winning time was.437 seconds. Round up.437 seconds to the next hundredth of a second. First, find the place you are rounding to:.437. The digit to the right is not, so add to the digit you are rounding to:.447. Finally, drop all digits to the right of hundredths:.44. The official winning time is.44 seconds..43 seconds is rounded up to.44 seconds..43 seconds is rounded up to.43 seconds because every digit to the right of the hundredths place is a. In this problem, rounding up does not change the number at all..43 equals.43. Rounding to the Nearest Place Rounding to the nearest place is sometimes like rounding up and sometimes like rounding down. To round a decimal to the nearest place, follow these steps: Step : Find the digit to the right of the place you are rounding to. Step 2: If that digit is 5 or more, round up. If that digit is less than 5, round down. Mr. Wilson is labeling the grocery shelves with unit prices so customers can compare the cost of items. To find a unit price, he divides the price by the quantity. Often, the quotient has more decimal places than are needed, so he rounds to the nearest cent (the nearest hundredth). $.23422 is rounded down to $.23. $3.89822 is rounded up to $3.9. $.865 is rounded up to $.87.. Round down to tenths. a..62 b. 36.592 c..95 2. Round up to tenths. a..62 b. 36.592 c..95 3. Round to the nearest tenth. a..62 b. 36.59 c..95 Check your answers on page 434. 46 forty-six
Percents A percent is another way to name a fraction or decimal. Percent means per hundred, or out of a hundred. So, % has the same meaning as the fraction and the decimal.. And 6% has 6 the same meaning as and.6. The statement 6% of students were absent means that 6 out of students were absent. This does not mean that there were exactly students and that 6 of them were absent. It does mean that for every students, 6 students were absent. A percent usually represents a percent of something. The something is the whole (or ONE, or %). In the statement, 6% of the students were absent, the whole is the total number of students in the school. The word percent comes from the Latin per centum: Per means for and centum means one hundred. There are 25 students in Esmond School. One winter day, 6% of the students were absent. How many students were absent that day? Think: 25 5 For every students, 6 were absent. So, for every 5 students ( 2 of ), 3 were absent ( 2 of 6). 6 6 3 5 students were absent that day. Percents are used in many ways in everyday life: Business: 5% off means that the price of an item will be reduced by 5 cents for every cents the item usually costs. Statistics: 55% voter turnout means that 55 out of every registered voters will actually vote. School: An 8% score on a spelling test means that a student scored 8 out of possible points for that test. One way to score 8% is to spell 8 words correctly out of. Another way to score 8% is to spell 8 words correctly out of. Probability: A 3% chance of showers means that for every days that have similar weather conditions, you would expect it to rain on 3 of the days. forty-seven 47
Fractions, Decimals, and Percents Percents may be used to rename both fractions and decimals. Percents are another way of naming fractions with a denominator of. 25 You can think of the fraction as 25 parts per hundred, or 25 out of, and write 25%. You can rename the fraction 5 as * 2 2, or 5 * 2, or 2%. 75 75% can be written as, or 3 4. Percents are another way of naming decimals in terms of hundredths. Since. can be written as, you can think of. 39 as %. And you can think of.39 as, or 39%. 58% means 58 hundredths, or.58. Percents can also be used to name the whole. out of can be written as the fraction, or hundredths. This is the same as whole, or %. Around 9, a symbol that could represent thousandths was created. It was called per mille and was written %o. For example, 7%o was 7 written to represent,. This symbol is not used very often today. 85 5 85%.85 5%.5 7. 5 2% 2 2 37.5% 3.375 The amounts shown in the pictures below can be written as 4, or 25%, or.25. 4 4 * 25 25 * 25 25 But 25.25. And 25%. So, 4 25 and and.25 and 25% all name the same amount. 4 25%.25 48 forty-eight
Finding a Percent of a Number Finding a percent of a number is a basic problem that comes up over and over again. For example: A backpack that regularly sells for $6 is on sale for 2% off. What is the sale price? The sales tax on food is 5%. What is the tax on $8 worth of groceries? A borrower pays % interest on a car loan. If the loan is $6,, how much is the interest? There are many different ways of finding the percent of a number. Use a Fraction Some percents are equivalent to easy fractions. For example, 25 25% is the same as, or 4. It is usually easier to find 25% of a number by thinking of 25% as. What is 25% of 48? 25 Think: 25% 4, so 25% of 48 is the same as 4 of 48. Divide 48 into 4 equal groups. Each group is 4 of 48, and each group has 2. So, 25% of 48 is 2. What is 2% of 6? 2 Think: 2% 5, so 2% of 6 is the same as 5 of 6. Divide 6 into 5 equal groups. Each group is 5 of 6, and each group has 2. So, 2% of 6 is 2. 4 2% OFF ALL CAMPING GEAR Some easy fractions and percents: 2 5 4 25 3 4 75 5 2 2 5 4 3 5 6 4 5 8 3 3 7 7 5% 25% 75% 2% 4% 6% 8% % 3% 7% 9 9 9% forty-nine 49
If a percent does not equal an easy fraction, you might find % first. What is 7% of 3? %, so % of 3 is the same as of 3. If you divide 3 into equal groups, there are 3 in each group. % of 3 is 3. Then 7% of 3 is 7 * 3. So, 7% of 3 2. Sometimes it is helpful to find % first. What is 3% of 6? %. % of 6 is of 6. If you divide 6 into equal groups, each group has 6. % of 6 is 6. Then 3% of 6 is 3 * 6. So, 3% of 6 8. Use Decimal Multiplication Finding a percent of a number is the same as multiplying the number by the percent. Usually, it s easiest to change the percent to a decimal and use a calculator. What is 35% of 55? 35 35%.35 Change the percent to a decimal. Multiply using a calculator. Key in:.35 55 Answer: 9.25 If your calculator has a percent as a decimal. key, you don t need to rename the To find 35% of 55, key in 35 55 on Calculator A. Or, key in 55 35 on Calculator B. 35% of 55 is 9.25. % of 55 means * 55 or. * 55. 7 7% of 55 means * 55 or.7 * 55. 35 35% of 55 means * 55 or.35 * 55. The word of in problems like these means multiplication. Solve.. A $6 backpack is on sale for 2% off. What is the sale price? 2. The sales tax on food is 5%. What is the tax on $8 worth of groceries? 3. A borrower pays % interest on a $6, car loan. How much is the interest? 5 fifty Check your answers on page 434.
Calculating a Discount A discount is an amount taken off a regular price; it s the amount you save. Stores may display the regular price and percent discount and let customers figure out the sale price. If the percent discount is equivalent to an easy fraction, then a good way to solve this kind of problem is by using the fraction. The list price for a desk lamp is $5, but it is on sale at a 2% discount (2% off the list price). What are the savings? 2 Change 2% to a fraction: 2% 5. Since 2% 5, the discount is 5 of $5. 5 of $5 means 5 * $5. 5 * $5 5 * $5 The discount is $. 5 * $5 * $5 5 $ a c For any fractions b and d, a b * c a d * c b. * d If the percent discount is not equivalent to an easy fraction, it s usually best to change the percent to a decimal first, and then to multiply with paper and pencil or a calculator. The list price for a radio is $45. The radio is sold at a 2% discount (2% off the list price). 2 What are the savings? (Reminder: 2% = =.2) Paper and pencil: 2 2% of 45 means * 45, or.2 * 45. 2 2% of $45 = 2 5 * $45 * $4 2 * $ 45 * $5 4 $5.4 Calculator: Key in:.2 45. Interpret the answer 5.4 as $5.4. The discount is $5.4. Stores sometimes offer 33% off sales. What they often mean is 3 off, but they do not want to write the discount as a fraction ( 3 ) or as a percent that includes a fraction (33 3 %). Solve.. The list price of a pair of jeans is $3. The jeans are being sold at a % discount. What are the savings? Check your answers on page 434. 2. Movies cost $9., but shows before 4 P.M. are 25% off. How much cheaper are the early shows? fifty-one 5
Finding the Whole in Percent Problems Sometimes you know a percent and how much it s worth, but you don t know what the ONE is. The sale price of a CD player is $2. It is on sale for 6% of its list price. What is the list price? This problem can be solved in different ways. Solution : Use fractions. 6 Find an easy fraction that is equivalent to 6%. 6% This means that 3 5 of the list price is $2. 3 5 Since 3 5 of the list price is $2, 5 of the list price is $4 ( $2 3 $4). Then 5 5 of the list price is 5 * $4 $2. Solution 2: Use percents. 6% is worth $2. 2 So, % is worth $2 ( $ 6 $2), and % is worth $2 ( * $2 $2). The list price is $2. A tea kettle is on sale for 8% of its list price. The sale price is $4. What is the list price? Solution : Use fractions. 8 8% 4 5 This means that 4 5 of the list price is $4. Since 4 5 of the list price is $4, 5 of the list price is $ ( $4 4 $). Then 5 5 of the list price is 5 * $ $5. Solution 2: Use percents. 8% is worth $4. So, % is worth $.5 ( $4 8 $.5), and % is worth $5 ( * $.5 $5). The list price is $5. 52 fifty-two
In Alaska, there are about 3, Native Americans. These Native Americans make up about 6% of Alaska s population. What is the population of Alaska? Use a % strategy. First find %. Then multiply by to get %. Use your calculator to divide 3, by 6: Key in: 3, 6 Answer: 6437.5 Multiply by : Key in: 6437.5 Answer: 64375 The total population of Alaska is 643,75, or about 65,. Solve.. A bicycle is on sale for 5% of the list price. The sale price is $. What is the list price? 3. In Canada, there are about 6 million children aged 4 and younger. These children make up about 9% of Canada s population. What is the population of Canada? Check your answers on page 435. 2. A cellular telephone is on sale for 4% of the list price. The sale price is $6. What is the list price? fifty-three 53
The Whole Circle One full turn of a circle can be divided in various ways. One way is to divide it into four equal parts: a quarter turn, a half turn, three-fourths (or three-quarters) of a turn, and a full turn. A full turn can also be broken into 36 or equal parts. A Circle Protractor There are 36 equally-spaced marks around the edge of the Circle Protractor. Each pair of side-by-side marks measures one degree ( ) of angle measure. That means there are 36 degrees (36 ) in the whole circle. The Percent Circle There are equally-spaced marks around the edge of the Percent Circle. The marks define thin, pie-shaped wedges. Each wedge contains one percent (%) of the total area inside the circle. There are wedges (%) inside the whole circle. 54 fifty-four