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HOMEWORK 04. Due Thursday, October 02
Exercise 4.1. Answer the following questions.
- What is a rational number?
- Which fraction is called proper? Which fraction is called improper? Give two examples of proper fractions and two examples of improper fractions. Explain.
- Which number is called a mixed number? Give two examples of a mixed number.
- What is a decimal number? What is a terminating decimal number? What is a repeating decimal number?
Exercise 4.2. Reduce the following fraction to its lowest terms. Determine
whether this number is proper fraction or not. If not, present it as a mixed
number.
• 26 ÷ 91;
• 135 ÷ 120;
• 148 ÷ 124;
Solution.
Exercise 4.3. Present each decimal number as a ratio of two integers. DO
NOT reduce the fraction into lowest terms.
- 0 .2345, 0.452 (terminating)
- 0 .51, 5. 2 39 (repeating)
Solution. 0.2345 = 2345 10000
Let a = 0. 51515151 · · · = 0.51. We multiply this number by 10, 100,
1000, etc. until we see how to get rid of the repeating part.
We get 10a = 5. 1515151 · · · = 5.15.
100 a = 51. 51515151 · · · = 51.51. We see that 100a and a have the same
digits after the decimal point. Then, 100a − a = 51. 51 − 0 .51, or 99a = 51.
Therefore, a =
Let a = 5. 239393939 · · · = 5. 2 39. Then, 10a = 52. 39393939 · · · = 52.39,
100 a = 523. 93939393 · · · = 523.93, 1000a = 5239. 39393939 · · · = 5239.39.
We can see that the numbers 10a and 1000a have the same digits after the
decimal point. Therefore, 1000a − 10 a = 5239. 39 − 52 .39, or 990a = 5178.
Thus, a =
Exercise 4.4. Evaluate the expression. Explain each step.
÷
Exercise 4.9. Perform the indicated operation. Simplify the answer when
possible.
√^125
Solution. First, we simplify all square roots:
Second, we gather all terms:
4 × 2 ×
3 + 7 × 5 ×
3 − 10 × 4 ×
Exercise 4.10. Rationalize the denominator.
√^20
√^4 ·^5
√^5
√^5
√^19
√^10
√^10
√^6
√^50
√^25
= √^5
= √^5
√^7
=^5
