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1.4 Operations with Rational Numbers

1.4 Operations with Rational Numbers. What You Will Learn. Rewrite Fractions as equivalent fractions Add and subtract fractions Multiply and divide fractions Add, subtract, multiply, and divide decimals. Rewriting Fractions.

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1.4 Operations with Rational Numbers

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  1. 1.4 Operations with Rational Numbers

  2. What You Will Learn Rewrite Fractions as equivalent fractions Add and subtract fractions Multiply and divide fractions Add, subtract, multiply, and divide decimals

  3. Rewriting Fractions A fraction is a number that is written as a quotient, with a numerator and a denominator. The terms fraction and rational number are related, but are not exactly the same. The term fraction refers to a number’s form, whereas the term rational number refers to its classification. For instance, the number 2 is a fraction when it is written as , but it is a rational number regardless of how it is written.

  4. Example 1 – Writing Fractions in Simplest Form Divide out GCF of 6 Divide out GCF of 7 Divide out GCF of 24

  5. Rewriting Fractions You can obtain an equivalent fraction by multiplying the numerator and denominator by the same nonzero number or by dividing the numerator and denominator by the same nonzero number.

  6. Rewriting Fractions Here are some examples. Fraction Equivalent Fraction Operation Divide numerator and denominator by 3. Equivalent Fractions

  7. Example 2 – Writing Equivalent Fractions Write an equivalent fraction with the indicated denominator. Solution:

  8. Adding and Subtracting Fractions Let a, b, and c be integers with c≠ 0. To add or subtract the fractions a/c and b/c, use the follow rules. With like Denominators: With unlike denominators: Rewrite the fractions so that they have like denominators. Then use the rule for adding and subtracting fractions with like denominators.

  9. Example 4 – Adding and Subtracting with Like Denominators Add numerators. Subtract numerators.

  10. Adding and Subtracting Fractions To find a like denominator for two or more fractions, find the least common multiple (or LCM) of their denominators. For instance, the LCM of the denominators of and is 24. To see this, consider all multiples of 8 (8, 16, 24, 32, 40, 48, . . .) and all multiples of 12 (12, 24, 36, 48, . . .). The numbers 24 and 48 are common multiples, and the number 24 is the smallest of the common multiples.

  11. Example 5 – Adding and Subtracting with Unlike Denominators

  12. Example 6 – Finding the Yardage for a Clothing Design • A designer uses yards of material to make a skirt and yards to make a shirt. Find the total amount of material required. • Solution

  13. Multiplying and Dividing Fractions The procedure for multiplying fractions is simpler than those for adding and subtracting fractions. Regardless of whether the fractions have like or unlike denominators, you can find the product of two fractions by multiplying the numerators and multiplying the denominators. Let a, b, c, and d be integers with b ≠ 0and d ≠ 0. Then the product of and is

  14. Example 7 – Multiplying Fractions Multiply numerators and denominators. Simplify. Product of two negatives is positive. Multiply numerators and denominators. Divide out common factor. Write in simplest form.

  15. Multiplying and Dividing Fractions The reciprocal or multiplicative inverse of a number is the number by which it must be multiplied to obtain 1. For instance, the reciprocal of 3 is because . Similarly, the reciprocal of is because To divide two fractions, multiply the first fraction by the reciprocal of the second fraction.

  16. Multiplying and Dividing Fractions Another way of saying this is “invert the divisor and multiply.” Let a, b, c, and d be integers with b ≠ 0and d ≠ 0. Then the product of and is

  17. Example 8 – Dividing Fractions Invert divisor and multiply. Multiply numerators and denominators. Divide out common factors. Write in simplest form.

  18. Example 9 – Finding the numbers of Calories Burned You decide to take a tennis class. You burn about 400 calories per hour playing tennis. In one week, you played tennis for hour on Tuesday, 2 hours on Wednesday, and on Thursday. How many total calories did you burn playing tennis during that week? What was your average number of calories burned per day playing tennis?

  19. Example 9 – Finding the numbers of Calories Burned Solution The total number of calories you burned playing tennis during the week was The average number of calories burned per day was cont’d

  20. Operations with Decimals Rational numbers can be represented as terminating or repeating decimals. Here are some examples. Terminating Decimals Repeating Decimals Note that bar notation is used to indicate the repeated digit (or digits) in the decimal notation.

  21. Operations with Decimals To round a decimal, use the following rules. Determine the number of digits of accuracy you wish to keep. The digit in the last position you keep is called the rounding digit, and the digit in the first position you discard is called the decision digit. If the decision digit is 5 or greater, round up by adding 1 to the rounding digit. If the decision digit is 4 or less, round down by leaving the rounding digit unchanged.

  22. Example 10 – Adding and Multiplying Decimals a. Add 0.583, 1.06, and 2.9104. b. Multiply –3.57 and 0.032. Solution a. To add decimals, align the decimal points and proceed as in integer addition.

  23. Example 10 – Adding and Multiplying Decimals cont’d Solution b. To multiply decimals, use integer multiplication and then place the decimal point (in the product) so that the number of decimal places equals the sum of the decimal places in the two factors. Two decimal places Three decimal places Five decimal places

  24. Example 11 – Dividing Decimals Divide 1.483 by 0.56. Round the answer to two decimal places. Solution To divide 1.483 by 0.56, convert the divisor to an integer by moving its decimal point to the right. Move the decimal point in the dividend an equal numberof places to the right.

  25. Example 11 – Dividing Decimals cont’d Place the decimal point in the quotient directly above the newdecimal point in the dividend and then divide as with integers.

  26. Example 11 – Dividing Decimals cont’d Rounded to two decimal places, the answer is 2.65. This answer can be written as where the symbol ≈ means is approximately equal to.

  27. Example 12 – Finding a Cell Phone Charge A cellular provider charges $5.35 for the first 200 text messages per month and $0.10 for each additional test message. • Find the cost of 263 text messages. • Can you save money by switching to a plan that allows unlimited text messages for $10 per month?

  28. Example 12 – Finding a Cell Phone Charge cont’d • Solution • You sent or received 63 text messages above 200. The cost of these is • ($0.10)(63) = $6.30 • So, your total charge for the month is • $5.35 + $6.30 = $11.65 • b. If you continue to send and receive this number of text messages each month, you can save money by switching to plan that allows unlimited text messages for $10 per month.

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