3.4.0: Exercises — Read — Invisible Network

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

Exercise $\PageIndex{1}$

4.598

Exercise $\PageIndex{2}$

$\sqrt{144}$

Exercise $\PageIndex{3}$

$\sqrt{131}$

For the following exercises, reduce the fraction to lowest terms

Exercise $\PageIndex{4}$

$\dfrac{8}{10}$

Exercise $\PageIndex{5}$

$\dfrac{30}{105}$

Exercise $\PageIndex{6}$

$\dfrac{36}{539}$

Exercise $\PageIndex{7}$

$\dfrac{231}{490}$

Exercise $\PageIndex{8}$

$\dfrac{750}{17,875}$

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

Exercise $\PageIndex{9}$

Convert $\frac{25}{6}$ to a mixed number.

Exercise $\PageIndex{10}$

Convert $\frac{240}{53}$ to a mixed number.

Exercise $\PageIndex{11}$

Convert $2 \frac{3}{8}$ to an improper fraction.

Exercise $\PageIndex{12}$

Convert $15 \frac{7}{30}$ to an improper fraction.

Exercise $\PageIndex{13}$

Convert $\frac{4}{9}$ to decimal form.

Exercise $\PageIndex{14}$

Convert $\frac{13}{20}$ to decimal form.

Exercise $\PageIndex{15}$

Convert $\frac{27}{625}$ to decimal form.

Exercise $\PageIndex{16}$

Convert $\frac{11}{14}$ to decimal form.

Exercise $\PageIndex{17}$

Convert 0.23 to fraction form and reduce to lowest terms.

Exercise $\PageIndex{18}$

Convert 3.8874 to fraction form and reduce to lowest terms.

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

Exercise $\PageIndex{19}$

$\frac{3}{5}+\frac{3}{10}$

Exercise $\PageIndex{20}$

$\frac{3}{14}+\frac{8}{21}$

Exercise $\PageIndex{21}$

$\frac{13}{36}-\frac{14}{99}$

Exercise $\PageIndex{22}$

$\frac{13}{24}-\frac{4}{117}$

Exercise $\PageIndex{23}$

$\frac{3}{7} \times \frac{21}{48}$

Exercise $\PageIndex{24}$

$\frac{48}{143} \times \frac{77}{120}$

Exercise $\PageIndex{25}$

$\frac{14}{27} \div \frac{7}{12}$

Exercise $\PageIndex{26}$

$\frac{44}{75} \div \frac{484}{285}$

Exercise $\PageIndex{27}$

$\left(\frac{3}{5}+\frac{2}{7}\right) \times \frac{10}{21}$

Exercise $\PageIndex{28}$

$\frac{3}{8} \times\left(\frac{13}{12}-\frac{35}{36}\right)$

Exercise $\PageIndex{29}$

$\left(\frac{3}{7}+\frac{5}{16}\right)^2-\frac{5}{12}$

Exercise $\PageIndex{30}$

$\frac{3}{8} \times\left(\frac{4}{9}-\frac{1}{8}\right)^2$

Exercise $\PageIndex{31}$

$\left(\frac{2}{5} \times\left(\frac{7}{8}-\frac{2}{3}\right)\right)^2 \div\left(\frac{4}{9}+\frac{5}{6}\right)+\frac{7}{12}$

Exercise $\PageIndex{32}$

$\left(\frac{1}{5} \div\left(\frac{3}{10}+\frac{11}{15}\right)\right) \times\left(\frac{2}{21}+\frac{5}{9}\right)-\left(\frac{8}{15} \div \frac{4}{33}\right)^2$

Exercise $\PageIndex{33}$

Find a rational number between $\frac{8}{17}$ and $\frac{15}{28}$

Exercise $\PageIndex{34}$

Find a rational number between $\frac{3}{50}$ and $\frac{13}{98}$.

Exercise $\PageIndex{35}$

Find two rational numbers between $\frac{3}{10}$ and $\frac{19}{45}$.

Exercise $\PageIndex{36}$

Find three rational numbers between $\frac{5}{12}$ and $\frac{175}{308}$.

Exercise $\PageIndex{37}$

Convert $24 \%$ to fraction form and reduce completely.

Exercise $\PageIndex{38}$

Convert 95\% to fraction form and reduce completely.

Exercise $\PageIndex{39}$

Convert 0.23 to a percentage.

Exercise $\PageIndex{40}$

Convert 1.22 to a percentage.

Exercise $\PageIndex{41}$

Determine 30\% of 250.

Exercise $\PageIndex{42}$

Determine $75 \%$ of 600.

Exercise $\PageIndex{43}$

If $25 \%$ of a group is 41 members, how many members total are in the group?

Exercise $\PageIndex{44}$

If $80 \%$ of the total is 60 , how much is in the total?

Exercise $\PageIndex{45}$

13 is what percent of 20 ?

Exercise $\PageIndex{46}$

80 is what percent of 320 ?

Exercise $\PageIndex{47}$

Professor Donalson's history of film class has 60 students. Of those students, $\frac{2}{5}$ say their favorite movie genre is comedy. How many of the students in Professor Donalson's class name comedy as their favorite movie genre?

Exercise $\PageIndex{48}$

Naia's dormitory floor has 80 residents. Of those, $\frac{3}{8}$ play Fortnight for at least 15 hours per week. How many students on Naia's floor play Fortnight at least 15 hours per week?

Exercise $\PageIndex{49}$

In Tara's town there are 24,000 people. Of those, $\frac{13}{100}$ are food insecure. How many people in Tara's town are food insecure?

Exercise $\PageIndex{50}$

Roughly $\frac{4}{5}$ of air is nitrogen. If an enclosure holds 2,000 liters of air, how many liters of nitrogen should be expected in the enclosure?

Exercise $\PageIndex{51}$

To make the dressing for coleslaw, Maddie needs to mix it with $\frac{3}{5}$ mayonnaise and $\frac{2}{5}$ apple cider vinegar. If Maddie wants to have 8 cups of dressing, how many cups of mayonnaise and how many cups of apple cider vinegar does Maddie need?

Exercise $\PageIndex{52}$

Malika is figuring out their schedule. They wish to spend $\frac{4}{15}$ of their time sleeping, $\frac{1}{3}$ of their time studying and going to class, $\frac{1}{5}$ of their time at work, and $\frac{2}{15}$ of their time doing other activities, such as entertainment or exercising. There are 168 hours in a week. How many hours in a week will Malika spend:

Sleeping?
Studying and going to class?
Not sleeping?

Exercise $\PageIndex{53}$

Roughly $20.9 \%$ of air is oxygen. How much oxygen is there in 200 liters of air?

Exercise $\PageIndex{54}$

$65 \%$ of college students graduate within 6 years of beginning college. A first-year cohort at a college contains 400 students. How many are expected to graduate within 6 years?

Exercise $\PageIndex{55}$

A $20 \%$ discount is offered on a new laptop. How much is the discount if the new laptop originally cost $\$ 700$ ?

Exercise $\PageIndex{56}$

Leya helped at a neighborhood sale and was paid $5 \%$ of the proceeds. If Leya is paid $\$ 171.25$, what were the total proceeds from the neighborhood sale?

Exercise $\PageIndex{57}$

Unit Conversion. 1 kilogram ( kg ) is equal to 2.20462 pounds. Convert 13 kg to pounds. Round to three decimal places, if necessary.

Exercise $\PageIndex{58}$

Unit Conversion. 1 kilogram (kg) is equal to 2.20462 pounds. Convert 200 pounds to kilograms. Round to three decimal places, if necessary.

Exercise $\PageIndex{59}$

Unit Conversion. There are 12 inches in a foot, 3 feet in a yard, and 1,760 yards in a mile. Convert 10 miles to inches. To do so, first convert miles to yards. Next, convert the yards to feet. Last, convert the feet to inches.

Exercise $\PageIndex{60}$

Unit Conversion. There are 1,000 meters ( m ) in a kilometer ( km ), and 100 centimeters ( cm ) in a meter. Convert 4 km to centimeters.

Exercise $\PageIndex{61}$

Markup. 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 $\$ 1.50$ per box and sells the cereal for $\$ 2.29$.

Determine the markup in dollars.
The markup is what percent of the original cost? Round the percentage to one decimal place.

Exercise $\PageIndex{62}$

In this exercise, we explore what happens when an item is marked up by a percentage, and then marked down using the same percentage.
Wegmans purchases an item for $\$ 5$ per unit. The markup on the item is $25 \%$.

Calculate the markup on the item, in dollars.
What is the price for which Wegmans sells the item? This is the price Wegmans paid, plus the markup.
Suppose Wegmans then offers a 25\% discount on the sale price of the item (found in part b). In dollars, how much is the discount?
Determine the price of the item after the discount (this is the sales price of the item minus the discount). Round to two decimal places.
Is the new price after the markup and discount equal to the price Wegmans paid for the item? Explain.

Exercise $\PageIndex{63}$

Repeated Discounts. In this exercise, we explore applying more than one discount to an item. Suppose a store cuts the price on an item by $50 \%$, and then offers a coupon for $25 \%$ off any sale item. We will find the price of the item after applying the sale price and the coupon discount.

The original price was $\$ 150$. After the $50 \%$ discount, what is the price of the item?
The coupon is applied to the discount price. The coupon is for $25 \%$. Find $25 \%$ of the sale price (found in part a).
Find the price after applying the coupon (this is the value from part a minus the value from part b).
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.
Determine the effective discount percentage, which is the total amount saved divided by the original price of the item.
Was the effective discount percentage equal to $75 \%$, which would be the $50 \%$ plus the $25 \%$ ? Explain.

Converting Repeating Decimals to a Fraction

It was mentioned in the section that repeating decimals are rational numbers. To convert a repeating decimal to a rational number, perform the following steps:

Step 1: Label the original number $S$.
Step 2: Count the number of digits, $n$, in the repeating part of the number.
Step 3: Multiply $S$ by $10^n$, and label this as $10^n \times S$.
Step 4: Determine $10^n-1$.
Step 5: Calculate $10^n \times S-S$. If done correctly, the repeating part of the number will cancel out.
Step 6: If the result from Step 5 has decimal digits, count the number of decimal digits in the number from Step 5. Label this $m$.
Step 7: Remove the decimal from the result of Step 5.
Step 8: Add $m$ zeros to the end of the number from Step 4.
Step 9: Divide the result from Step 7 by the result from Step 8. This is the fraction form of the repeating decimal.

Exercise $\PageIndex{64}$

Convert $0 . \overline{7}$ to fraction form.

Exercise $\PageIndex{65}$

Convert $0 . \overline{45}$ to fraction form.

Exercise $\PageIndex{66}$

Convert $3.1 \overline{5}$ to fraction form.

3.4.0: Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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Converting Repeating Decimals to a Fraction

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