1.3E Exercises

Place Value & Rounding

Consider the number 2024.05041339 and answer the following questions. (I made this number up because I wrote this question at 13:39 on 05/04/2024 lol.)

1. What digit is in the tens place?
2. What digit is in the hundreds place?
3. What digit is in the tenths place?
4. What digit is in the hundredths place?
5. What digit is in the thousandths place?
6. Round to the nearest integer.
7. Round to the nearest tenth (round to one decimal place).
8. Round to the nearest hundredth (two decimal places).
9. Round to five decimal places.
10. Round to seven decimal places.

Answer

1. 2
2. 0
3. 0
4. 5
5. 0
6. 2024
7. 2024.1
8. 2024.05
9. 2024.05041
10. 2024.0504134

Decimal Arithmetic

Compute the following using mental math for practice.

1. $3(0.\overline{3})$
2. $0.25 \cdot 4$
3. $0.25 + 0.5$
4. $0.2 + 0.3$
5. $1 - 0.5$
6. $0.5 - 1$
7. $2(0.4)$
8. $3(0.1)$
9. $2(0.6)$
10. $2(0.9)$
11. $2.4(100)$
12. $3.35 \cdot 1000$
13. $\frac{2.4}{100}$
14. $\frac{14.8}{10}$

Answer

1. That's one third times three, so 1.
2. 1
3. 0.75
4. 0.5
5. 0.5
6. -0.5
7. 0.8
8. 0.3
9. 1.2 (12 with the point moved one place in)
10. 1.8
11. 240
12. 3350
13. 0.024
14. 1.48

Converting Between Decimals & Fractions

1. Write $\frac{1}{3}$ as a decimal. Then round to four decimal places.
2. Write $\frac{2}{3}$ as a decimal. Then round to four decimal places.
3. Write $2.5$ as a fraction.
4. Write $-1.75$ as a fraction
5. Write the mixed number $3\frac{1}{5}$ as a decimal.
6. Write the mixed number $4\frac{1}{2}$
7. Write $-0.125$ as a fraction.
8. Write $0.42$ as a fraction.

Answer

1. $0.33333... \approx 0.3333$
2. $0.66666... \approx 0.6667$
3. That's 2 and a half, so $2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}$
4. $-\frac{7}{4}$
5. $3.2$
6. $4.5$
7. $- \frac{1}{8}$
8. $\frac{42}{100} = \frac{21}{50}$

Percents

1. Convert 75% to a decimal.
2. Convert 110% to a decimal.
3. What is 50% of 24?
4. What is 10% of 300?

Answer

1. 0.75
2. 1.10
3. 12
4. 30

Calculating Tip

You should tip 20% and your bill is $12.50, and your phone is dead so you can't get a calculator out. Luckily, you know you can find 10% of 15.50 in your head and then just double it! What is the tip amount? What is the bill total with tip?

Answer

The tip is $2.50, so your total bill will be $15.

(Challenge) Finding a Rational Number From Repeating Decimal

Let's figure out how to write $0.123123123123... = 0.\overline{123}$ as a fraction $\frac{m}{n}$, where $m$ and $n$ are integers.

1. First nickname the decimal as $x = 0.123123123123...$ and notice that its repeating pattern is three digits long.
2. Since the pattern is three digits long, we will multiply $x$ by the third power of 10, so compute $1000x$ using the decimal moving trick.
3. Now subtract $1000x - x = ?$.
4. We'll see more about factoring later, but $1000x - x = (1000-1)x = 999x$. Using this fact, we can write $x = \frac{?}{999}$. That's a rational number, technically!

Answer

1. Duly noted.
2. $1000x = 1000(0.123123123...) = 123.123123123...$
3. $1000x - x = 123.123123123... - 0.123123123...$ calculated the vertical way looks like this:
4. So $999x = 123$ and dividing both sides by $999$ gives $x = \frac{123}{999}$, a ratio of two integers!

Gas Station Games

This is a true story. The other day, I was putting gas in my car, and I managed to stop the pump at exactly 5.00 gallons and exactly $18.00 total cost. Find (a) the price per gallon at that particular Stewart's on that particular day.

It made me curious about what other prices lead to nice whole number gallons and cost possibilities... (b) Investigate this idea and see if you can figure out what integer number of gallons I would need to get to have my total cost come out to an integer, if the price is $2.75 /gal. And (c) is it possible for me to win this "game" if the price per gallon is $3.239? I can only get up to 9 whole gallons of gas at a time, because my car has a 10-gallon tank.

Answer

(a) We can find the price per gallon by just dividing the total cost by the number of gallons I got. That is, $\dfrac{$18}{5}$ is $3$ and $\frac{3}{5}$. I can write that as a decimal as $3.60 per gallon. (Gas stations try to make you feel better by saying the price is $3.599, but it's all money so it's rounded to two decimal places anyway, lol.)

(b) We just computed the price by dividing the gallons by the cost. In letters, we could nickname price $p$, number of gallons $n$, and total cost $m$. The whole point of the game is to have $p = \dfrac{m}{n}$ with the $m$ and $n$ integers. That means we're just looking for the rational number $p$. Or, we're looking for a way to write the price (given to us in a decimal) as rational number so we can see the integer top and integer bottom! If the price is $2.75 per gallon, we convert from two-and-three-fourths to $\dfrac{11}{4}$. The top integer is $m$, the total cost, and the bottom integer is $n$, the total gallons. So at the price $2.75, I can get exactly 4.00 gallons for exactly $11.00.

(c) I don't know what fraction $3.239$ or even $3.24$ is equivalent to off the top of my head. You can always figure it out, though, by writing

$$3.24 = 3 + \frac{24}{100} = \frac{324}{100} \notag$$

and trying to cancel/simplify that fraction to some nicer integers. I first say, 4 goes into both the top and bottom, so I cancel that away.

$$\frac{324}{100} = \frac{ 4\cdot 81}{4\cdot 25} = \frac{81}{25} \notag$$

At this point, there's no more cancelling I can do, as the top and bottom have no more factors in common. So my only option for whole number gallons and cost would be to buy 25.00 gallons for $81.00, but I can't put that much in my car, so if the price is $3.239 I can't win the game. :(