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In the summer months, a common problem people suffer from is dehydration – not enough water in their body. Waiting until you feel thirsty is not a good indicator that it’s time to drink water. In fact, once thirst has settled in, you are already dehydrated! A general guideline is that people should drink eight 8-ounce glasses of water per day. According to this guideline, how many gallons of water should a person drink in a day, given that 1 gallon = 4 quarts and 1 quart = 32 ounces? Express your answer as a decimal to the nearest tenth.
Eight 8-ounce glasses of water is 8 × 8 = 64 ounces of water. Since it takes exactly double that amount, or 128 ounces, to make a gallon, the guideline for water consumption is 0.5 gallons of water per day.
When dehydrated, your body needs water. Juices and sodas are not the optimal beverages for replenishing your body with the water it needs. And if you take cost into account, you’ll definitely want to grab a glass of water! It is estimated that 4000 glasses of tap water cost the same as a six-pack of soda. If a six-pack of soda costs $2.99, how many glasses of water would have a cost of 10¢? Express your answer to the nearest whole number.
If a six-pack of soda costs $2.99, then a glass of water would cost 2.99 ÷ 4000 = 0.0007475 cents. To determine how many glasses have a cost of 10 cents, we multiply the ratio of glasses to cents by 0.10 to get (1/0.0007475)(0.10) ≈ 133.779. So, the number of glasses of water that have a cost of 10 cents is about 134 glasses of water.
Lake Tahoe is the second deepest lake in the U.S. and it holds 40 trillion gallons of water – enough to cover the state of California to a depth of 14 inches! Given that 1 ft3 = 7.48 gallons and 1 mile = 5280 ft, how many square miles are in the area of California? Express your answer to the nearest thousand.
First, 40 trillion gallons of water is equivalent to [(40 × 1012)/7.48] = [(10 × 1012)/1.87] ft3 of water. Next, we are told that this water would have depth 14 inches × 1/12 = 7/6 feet. At that depth, [(10 × 1012)/1.87] ft3 of water would cover [((10 × 1012)/1.87)/(7/6)] = [((10 × 1012)/1.87)(6/7)] ft2. Finally, since (1 mi)2 = (5280 ft)2, it follows that the number of square miles covered would be [((10 × 1012)/1.87)(6/7)]/(5280)2 ≈ 164,415.8790 mi2, which means that the area of California is about 164,000 mi2.
♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦
A town in South Dakota decides to have an “End of the Summer” pool party and invites all of the households in the town that have school aged children.
Twenty-seven households are invited to the party and the ratio of 5-person households to 4‑person households to 3-person households to 2-person households invited is 1:4:3:1, respectively. (The invite list did not include households of any other sizes.) How many people from households with less than 4 people are invited to the party?
To determine how many households of each size are invited, first set up a proportion for each of the household sizes that are smaller than 4. For two-person households, we have 1/9 = x/27, and for three‑person households, we have 3/9 = y/27. Now cross multiply and divide each proportion to get 9x = 27 → x = 3 two-person households are invited, and 9y = 81 → y = 9 three-person households were invited. Now we’ll multiply the number of two- and three-person households each by the number of people in that size household, and add the results: (3)(2) + (9)(3) = 6 + 27 = 33 people from households with less than four people are invited.
On the day of the pool party, 75% of the invited people attend. If 1 out of every 12 of the attendees brings one uninvited guest each, how many people attend the pool party?
Set up proportions, cross multiply and divide to determine the number of households of each size that were invited. We have 4/9 = a/27 → 9a = 108 → a = 12 four-person households were invited, and 1/9 = b/27 → 9b = 27 → b = 3 five-person households were invited. From the last question, we know that 3 two-person households were invited and 6 three-person households were invited. Now multiply the number of households of each size by the number of people in that size household and add the results to get (3)(2) + (9)(3) + (12)(4) + (3)(5) = 6 + 27 + 48 + 15 = 96 people were invited. Multiply the number of people invited by the decimal form of the percentage of invitees that attended and we see that (96)(0.75) = 72 of the invited people attend. Setting up a proportion to determine how many of the attendees brought an uninvited guest, we get 1/12 = c/72. When we cross-multiply and divide, we find that 12c = 72 → b = 6 people each bring 1 uninvited person, so a total of 6 uninvited guests are in attendance. Add the number of uninvited guests to the number of invited guests and we find the total number of people in attendance at the pool party was 72 + 6 = 78 people.
The party planning committee planned for only 70 people to attend, and it purchased enough supplies to feed each attendee two ¼-lb hamburgers. Because they underestimated the number of attendees, how many additional pounds of meat must be purchased so that everyone receives two ¼-lb hamburgers?
Multiply the number of hamburgers each guest ate by the weight of each hamburger, in pounds, to get (2)(1/4) = 1/2 lb of hamburger per person. There are 78 – 70 = 8 more attendees than had been planned for. Multiply the number of pounds of hamburger each guest gets by the number of extra people and we see that (1/2)(8) = 4 lbs of additional meat are needed.
At the beginning of the pool party, the rectangular pool, which is 50 ft by 25 ft with an average depth of 5 ft, was completely filled with water. By the end of the day, the water level had lowered by 6 in. By what percent had the amount of water in the pool decreased over the course of the day?
Multiply the length, width, and depth of the pool to calculate the volume of water originally in the pool, in cubic feet, to get (50)(25)(5) = 6250 ft3. Multiply the length, width, and the number of feet the depth decreased by to calculate the volume of water lost throughout the day. We get (50)(25)(.5) = 625 ft3. Divide the amount of water lost by the amount of water originally in the pool and multiply by 100 to calculate the percentage of water lost and the result is (625/6250)(100) = 10% of the water was lost.
♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦
The date of the beginning of the school year has long been a topic of debate in many areas across the United States, but the trend is leaning towards starting school earlier rather than later. Basically, in the last 10 years, the percent of public schools going back to school before Labor Day has increased by 63%. It is now estimated that 75% of public schools are back in session before Labor Day, rather than after. What percent of public schools were back in session before Labor Day ten years ago, to the nearest whole percent?
We are looking for the initial percentage before a 63% increase raised it to 75%. If we increase a number by 63%, we are essentially multiplying it by 1.63. So, we have an unknown “x”, which is multiplied by 1.63 and becomes .75. Our equation looks like 1.63x = .75. Dividing both sides by 1.63 shows us that x = .4601227 or 46%, to the nearest whole percent.
School systems around the country are dealing with major teacher shortages and have not been able to find a remedy for the problem. Suppose Lincoln Middle School had 483 students and averaged 23 students per teacher during the 2022-2023 school year. If Lincoln Middle School’s student population increases by 14% this year and 10% of last year’s teachers leave, how many new teachers will Lincoln need to hire to keep its average of 23 students per teacher? Express your answer to the nearest whole number.
First, let’s find out how many teachers Lincoln Middle School had during the 2022-2023 school year. Dividing 483 by 23, we see that there were 21 teachers last year. This year, there is an increase of 14% for its student population, which means there are 483(1.14) = 550.62 or 551 students. We want to keep the same student-to-teacher ratio, so dividing 551 by 23, we see that we will need 23.956522 or 24 teachers for this school year. However, we need to remember that we lost 10% of the teachers from last year, so we only have 90% of them returning, which is .90(21) = 18.9 or 19 teachers. Therefore, Lincoln Middle School will need 24 – 19 = 5 new teachers for the 2023-2024 school year.
It’s the first day of school and you are excited that you have the same 1st period class (Introduction to Theater) as your best friend. You both walk into the classroom and see that you are the first two students to arrive and that there are 20 chairs arranged in a circle (numbered 1-20, in order, with chair #20 next to chair #1). The teacher announces that there are exactly 20 students in the class, and she will be randomly assigning a seat to each student. To do this, she will start with the first student to arrive to class and pull a number (1-20) from a hat to determine which seat the student should sit in. Then she will do the same for the second student to arrive, the third student, and so on until each student has been assigned to a seat. What is the probability that you will be sitting next to your best friend? Express your answer as a common fraction.
Since you and your friend were the first two students to arrive to the classroom, we know that all of the seats are empty as the two of you are getting your seat assignments. Since you are the first to be seated, you could be placed in any of the chairs. So, there are 20 different seats you could get. The first scenario would be that you are placed in seat #1 (which is a 1/20 chance). Then, your friend would need to get placed in seat #2 or seat #20 (which is a 2/19 chance). Therefore, that scenario has a (1/20)(2/19) = 1/190 chance of happening. The second scenario would be that you get placed in seat #2 (which is a 1/20 chance). Then, your friend would need to get placed in seat #3 or seat #1 (which is a 2/19 chance). Therefore, this second scenario has a (1/20)(2/19) = 1/190 chance of happening. Notice that there are 18 more scenarios (since there are 18 more seats that you could be assigned to), and each scenario has a 1/190 chance of happening. Therefore, the probability of you and your friend being seated next to each other is 20(1/190) = 2/19, as a common fraction.
Another way of looking at it: Wherever you are seated, there will be 19 seats left, two of which will be next to you. So, your placement really doesn’t matter, and your friend will have a 2/19 chance of being placed next to you.
♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦
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In the summer months, a common problem people suffer from is dehydration – not enough water in their body. Waiting until you feel thirsty is not a good indicator that it’s time to drink water. In fact, once thirst has settled in, you are already dehydrated! A general guideline is that people should drink eight 8-ounce glasses of water per day. According to this guideline, how many gallons of water should a person drink in a day, given that 1 gallon = 4 quarts and 1 quart = 32 ounces? Express your answer as a decimal to the nearest tenth.
Eight 8-ounce glasses of water is 8 × 8 = 64 ounces of water. Since it takes exactly double that amount, or 128 ounces, to make a gallon, the guideline for water consumption is 0.5 gallons of water per day.
When dehydrated, your body needs water. Juices and sodas are not the optimal beverages for replenishing your body with the water it needs. And if you take cost into account, you’ll definitely want to grab a glass of water! It is estimated that 4000 glasses of tap water cost the same as a six-pack of soda. If a six-pack of soda costs $2.99, how many glasses of water would have a cost of 10¢? Express your answer to the nearest whole number.
If a six-pack of soda costs $2.99, then a glass of water would cost 2.99 ÷ 4000 = 0.0007475 cents. To determine how many glasses have a cost of 10 cents, we multiply the ratio of glasses to cents by 0.10 to get (1/0.0007475)(0.10) ≈ 133.779. So, the number of glasses of water that have a cost of 10 cents is about 134 glasses of water.
Lake Tahoe is the second deepest lake in the U.S. and it holds 40 trillion gallons of water – enough to cover the state of California to a depth of 14 inches! Given that 1 ft3 = 7.48 gallons and 1 mile = 5280 ft, how many square miles are in the area of California? Express your answer to the nearest thousand.
First, 40 trillion gallons of water is equivalent to [(40 × 1012)/7.48] = [(10 × 1012)/1.87] ft3 of water. Next, we are told that this water would have depth 14 inches × 1/12 = 7/6 feet. At that depth, [(10 × 1012)/1.87] ft3 of water would cover [((10 × 1012)/1.87)/(7/6)] = [((10 × 1012)/1.87)(6/7)] ft2. Finally, since (1 mi)2 = (5280 ft)2, it follows that the number of square miles covered would be [((10 × 1012)/1.87)(6/7)]/(5280)2 ≈ 164,415.8790 mi2, which means that the area of California is about 164,000 mi2.
♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦
A town in South Dakota decides to have an “End of the Summer” pool party and invites all of the households in the town that have school aged children.
Twenty-seven households are invited to the party and the ratio of 5-person households to 4‑person households to 3-person households to 2-person households invited is 1:4:3:1, respectively. (The invite list did not include households of any other sizes.) How many people from households with less than 4 people are invited to the party?
To determine how many households of each size are invited, first set up a proportion for each of the household sizes that are smaller than 4. For two-person households, we have 1/9 = x/27, and for three‑person households, we have 3/9 = y/27. Now cross multiply and divide each proportion to get 9x = 27 → x = 3 two-person households are invited, and 9y = 81 → y = 9 three-person households were invited. Now we’ll multiply the number of two- and three-person households each by the number of people in that size household, and add the results: (3)(2) + (9)(3) = 6 + 27 = 33 people from households with less than four people are invited.
On the day of the pool party, 75% of the invited people attend. If 1 out of every 12 of the attendees brings one uninvited guest each, how many people attend the pool party?
Set up proportions, cross multiply and divide to determine the number of households of each size that were invited. We have 4/9 = a/27 → 9a = 108 → a = 12 four-person households were invited, and 1/9 = b/27 → 9b = 27 → b = 3 five-person households were invited. From the last question, we know that 3 two-person households were invited and 6 three-person households were invited. Now multiply the number of households of each size by the number of people in that size household and add the results to get (3)(2) + (9)(3) + (12)(4) + (3)(5) = 6 + 27 + 48 + 15 = 96 people were invited. Multiply the number of people invited by the decimal form of the percentage of invitees that attended and we see that (96)(0.75) = 72 of the invited people attend. Setting up a proportion to determine how many of the attendees brought an uninvited guest, we get 1/12 = c/72. When we cross-multiply and divide, we find that 12c = 72 → b = 6 people each bring 1 uninvited person, so a total of 6 uninvited guests are in attendance. Add the number of uninvited guests to the number of invited guests and we find the total number of people in attendance at the pool party was 72 + 6 = 78 people.
The party planning committee planned for only 70 people to attend, and it purchased enough supplies to feed each attendee two ¼-lb hamburgers. Because they underestimated the number of attendees, how many additional pounds of meat must be purchased so that everyone receives two ¼-lb hamburgers?
Multiply the number of hamburgers each guest ate by the weight of each hamburger, in pounds, to get (2)(1/4) = 1/2 lb of hamburger per person. There are 78 – 70 = 8 more attendees than had been planned for. Multiply the number of pounds of hamburger each guest gets by the number of extra people and we see that (1/2)(8) = 4 lbs of additional meat are needed.
At the beginning of the pool party, the rectangular pool, which is 50 ft by 25 ft with an average depth of 5 ft, was completely filled with water. By the end of the day, the water level had lowered by 6 in. By what percent had the amount of water in the pool decreased over the course of the day?
Multiply the length, width, and depth of the pool to calculate the volume of water originally in the pool, in cubic feet, to get (50)(25)(5) = 6250 ft3. Multiply the length, width, and the number of feet the depth decreased by to calculate the volume of water lost throughout the day. We get (50)(25)(.5) = 625 ft3. Divide the amount of water lost by the amount of water originally in the pool and multiply by 100 to calculate the percentage of water lost and the result is (625/6250)(100) = 10% of the water was lost.
♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦
The date of the beginning of the school year has long been a topic of debate in many areas across the United States, but the trend is leaning towards starting school earlier rather than later. Basically, in the last 10 years, the percent of public schools going back to school before Labor Day has increased by 63%. It is now estimated that 75% of public schools are back in session before Labor Day, rather than after. What percent of public schools were back in session before Labor Day ten years ago, to the nearest whole percent?
We are looking for the initial percentage before a 63% increase raised it to 75%. If we increase a number by 63%, we are essentially multiplying it by 1.63. So, we have an unknown “x”, which is multiplied by 1.63 and becomes .75. Our equation looks like 1.63x = .75. Dividing both sides by 1.63 shows us that x = .4601227 or 46%, to the nearest whole percent.
School systems around the country are dealing with major teacher shortages and have not been able to find a remedy for the problem. Suppose Lincoln Middle School had 483 students and averaged 23 students per teacher during the 2022-2023 school year. If Lincoln Middle School’s student population increases by 14% this year and 10% of last year’s teachers leave, how many new teachers will Lincoln need to hire to keep its average of 23 students per teacher? Express your answer to the nearest whole number.
First, let’s find out how many teachers Lincoln Middle School had during the 2022-2023 school year. Dividing 483 by 23, we see that there were 21 teachers last year. This year, there is an increase of 14% for its student population, which means there are 483(1.14) = 550.62 or 551 students. We want to keep the same student-to-teacher ratio, so dividing 551 by 23, we see that we will need 23.956522 or 24 teachers for this school year. However, we need to remember that we lost 10% of the teachers from last year, so we only have 90% of them returning, which is .90(21) = 18.9 or 19 teachers. Therefore, Lincoln Middle School will need 24 – 19 = 5 new teachers for the 2023-2024 school year.
It’s the first day of school and you are excited that you have the same 1st period class (Introduction to Theater) as your best friend. You both walk into the classroom and see that you are the first two students to arrive and that there are 20 chairs arranged in a circle (numbered 1-20, in order, with chair #20 next to chair #1). The teacher announces that there are exactly 20 students in the class, and she will be randomly assigning a seat to each student. To do this, she will start with the first student to arrive to class and pull a number (1-20) from a hat to determine which seat the student should sit in. Then she will do the same for the second student to arrive, the third student, and so on until each student has been assigned to a seat. What is the probability that you will be sitting next to your best friend? Express your answer as a common fraction.
Since you and your friend were the first two students to arrive to the classroom, we know that all of the seats are empty as the two of you are getting your seat assignments. Since you are the first to be seated, you could be placed in any of the chairs. So, there are 20 different seats you could get. The first scenario would be that you are placed in seat #1 (which is a 1/20 chance). Then, your friend would need to get placed in seat #2 or seat #20 (which is a 2/19 chance). Therefore, that scenario has a (1/20)(2/19) = 1/190 chance of happening. The second scenario would be that you get placed in seat #2 (which is a 1/20 chance). Then, your friend would need to get placed in seat #3 or seat #1 (which is a 2/19 chance). Therefore, this second scenario has a (1/20)(2/19) = 1/190 chance of happening. Notice that there are 18 more scenarios (since there are 18 more seats that you could be assigned to), and each scenario has a 1/190 chance of happening. Therefore, the probability of you and your friend being seated next to each other is 20(1/190) = 2/19, as a common fraction.
Another way of looking at it: Wherever you are seated, there will be 19 seats left, two of which will be next to you. So, your placement really doesn’t matter, and your friend will have a 2/19 chance of being placed next to you.
♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦
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