Rational Numbers

For the following exercises, identify which of the following are rational numbers.

1. [latex]4.598[/latex]
2. [latex]\sqrt{131}[/latex]

For the following exercises, reduce the fraction to lowest terms

3. [latex]\frac{30}{105}[/latex]
4. [latex]\frac{231}{490}[/latex]

For the following exercises, do the indicated conversion. If it is a repeating decimal, use the correct notation.

5. Convert [latex]\frac{25}{6}[/latex] to a mixed number.
6. Convert [latex]2\frac{3}{8}[/latex] to an improper fraction.
7. Convert [latex]\frac{4}{9}[/latex] to decimal form.
8. Convert [latex]\frac{27}{625}[/latex] to decimal form.
9. Convert [latex]0.23[/latex] to fraction form and reduce to lowest terms.

For the following exercises, perform the indicated operations. Reduce to lowest terms.

10. [latex]\frac{3}{5} + \frac{3}{10}[/latex]
11. [latex]\frac{13}{36} - \frac{14}{99}[/latex]
12. [latex]\frac{3}{7} \times \frac{21}{48}[/latex]
13. [latex]\frac{14}{27} \div \frac{7}{12}[/latex]
14. [latex](\frac{3}{5} + \frac{2}{7}) \times \frac{10}{21}[/latex]
15. [latex](\frac{3}{7} + \frac{5}{16})^2 - \frac{5}{12}[/latex]
16. [latex](\frac{2}{5} \times (\frac{7}{8} - \frac{2}{3}))^2 \div (\frac{4}{9} + \frac{5}{6}) + \frac{7}{12}[/latex]

For the following exercises, solve the given question.

17. Convert [latex]24\%[/latex] to fraction form and reduce completely.
18. Convert [latex]0.23[/latex] to a percentage.
19. Determine [latex]30\%[/latex] of [latex]250[/latex]
20. If [latex]25\%[/latex] of a group is [latex]41[/latex] members, how many members total are in the group?
21. [latex]13[/latex] is what percent of [latex]20[/latex]?
22. Professor Donalson’s history of film class has [latex]60[/latex] students. Of those students, [latex]\frac{2}{5}[/latex] say their favorite movie genre is comedy. How many of the students in Professor Donalson’s class name comedy as their favorite movie genre?
23. In Tara’s town there are [latex]24[/latex],[latex]000[/latex] people. Of those, [latex]\frac{13}{100}[/latex] are food insecure. How many people in Tara’s town are food insecure?
24. To make the dressing for coleslaw, Maddie needs to mix it with [latex]\frac{3}{5}[/latex] mayonnaise and [latex]\frac{2}{5}[/latex] apple cider vinegar. If Maddie wants to have [latex]8[/latex] cups of dressing, how many cups of mayonnaise and how many cups of apple cider vinegar does Maddie need?
25. Roughly [latex]20.9\%[/latex] of air is oxygen. How much oxygen is there in [latex]200[/latex] liters of air?
26. A [latex]20\%[/latex] discount is offered on a new laptop. How much is the discount if the new laptop originally cost [latex]$700[/latex]?
27. [latex]1[/latex] kilogram (kg) is equal to [latex]2.20462[/latex] pounds. Convert [latex]13[/latex] kg to pounds. Round to three decimal places, if necessary.
28. There are [latex]12[/latex] inches in a foot, [latex]3[/latex] feet in a yard, and [latex]1,760[/latex] yards in a mile. Convert [latex]10[/latex] miles to inches. To do so, first convert miles to yards. Next, convert the yards to feet. Last, convert the feet to inches.
29. In this exercise, we introduce the concept of markup. The markup on an item is the difference between how much a store sells an item for and how much the store paid for the item. Suppose Wegmans (a northeastern U.S. grocery chain) buys cereal at [latex]$1.50[/latex] per box and sells the cereal for [latex]$2.29[/latex].
    1. Determine the markup in dollars.
    2. The markup is what percent of the original cost? Round the percentage to one decimal place.
30. In this exercise, we explore applying more than one discount to an item. Suppose a store cuts the price on an item by [latex]50\%[/latex], and then offers a coupon for [latex]25\%[/latex] off any sale item. We will find the price of the item after applying the sale price and the coupon discount.
    1. The original price was [latex]$150[/latex]. After the [latex]50\%[/latex] discount, what is the price of the item?
    2. The coupon is applied to the discount price. The coupon is for [latex]25\%[/latex]. Find [latex]25\%[/latex] of the sale price (found in part a).
    3. Find the price after applying the coupon (this is the value from part a minus the value from part b).
    4. The total amount saved on the item is the original price after all the discounts. Determine the total amount saved by subtracting the final price paid (part c) from the original price of the item.
    5. Determine the effective discount percentage, which is the total amount saved divided by the original price of the item.
    6. Was the effective discount percentage equal to [latex]75\%[/latex], which would be the [latex]50\%[/latex] plus the [latex]25\%[/latex]? Explain.

Irrational Numbers

1. Identify which of the following numbers are irrational: [latex]\sqrt{441}[/latex], [latex]4.33[/latex], [latex]\sqrt{70}[/latex], [latex]5 + 9\pi[/latex]

For the following exercises, simplify the square root by expressing it in lowest terms.

2. [latex]\sqrt{12}[/latex]
3. [latex]\sqrt{605}[/latex]
4. [latex]\sqrt{112}[/latex]
5. [latex]\sqrt{2,940}[/latex]
6. [latex]\sqrt{3,240}[/latex]

For the following exercises, perform the arithmetic operations without a calculator, if possible. If it is not possible, state why.

7. [latex]4\sqrt{3} + 2\sqrt{3}[/latex]
8. [latex]9\sqrt{7} - 16\sqrt{7}[/latex]
9. [latex]8\pi - 13\sqrt{2}[/latex]
10. [latex]7.2\pi + 8.6\pi[/latex]
11. [latex]19.8\sqrt{12} - 6.1\sqrt{3}[/latex]
12. [latex]4\sqrt{15} \times 3\sqrt{10}[/latex]
13. [latex]4.5\sqrt{154} \div 3\sqrt{77}[/latex]

Real Numbers

For the following exercises, identify each number as a natural number, an integer, a rational number, an irrational number, or a real number.

1. [latex]\frac{1}{3}[/latex]
2. [latex]\sqrt{19}[/latex]
3. [latex]\pi[/latex]
4. [latex]13[/latex]

For the following exercises, identify the property of real numbers that is being illustrated.

5. [latex]14 + 38.9 = 38.9 + 14[/latex]
6. [latex]38 \times 16 = 16 \times 38[/latex]
7. [latex]4 + (8 + \sqrt{7}) = (4 + 8) + \sqrt{7}[/latex]
8. [latex](5.6 \times 8.7) \times 6=5.6 \times (8.7 \times 6)[/latex]
9. [latex]4 \times (5 + \sqrt{3}) = 4 \times 5 + 4 \times \sqrt{3}[/latex]
10. [latex]-3.4\pi \times ( -\frac{1}{3.4\pi}) = 1[/latex]
11. [latex]4\sqrt{35} + 0 = 4\sqrt{35}[/latex]
12. [latex](−10\pi) + 10\pi = 0[/latex]
13. [latex]3\pi + 7\sqrt{21} = 7\sqrt{21} + 3\pi[/latex]

For the following exercises, use properties of real numbers and mental math to calculate the expression.

14. [latex]43+62+17[/latex]
15. [latex]5 \times 13 \times 4[/latex]
16. [latex]46+77+23+24+103[/latex]
17. [latex]4 \times 13 \times 25[/latex]
18. [latex]13 \times 99[/latex]
19. [latex]43 \times 101[/latex]