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Luis plans to purchase some new baseball bats and gloves before spring training begins. Luis wants to buy the same bats and gloves he purchased last spring, but he can’t remember the price of each item. Luis recalls making two purchases last spring, each totaling $135 and both tax exempt. The first purchase was for a glove and three bats. The second purchase was for two gloves and a bat. If the current prices of these items are the same as last spring, how much will Luis pay this spring for three gloves and four bats?
If we let g and b represent the prices of a glove and a bat, respectively, we can derive two equations: g + 3b = 135 and 2g + b = 135. We want to determine the price of three gloves and four bats. Combined, the two purchases from last spring were for (g + 3b) + (2g + b) = 3g + 4b, which is three gloves and four bats. So, we can add the two totals to find that the price is 135 + 135 = $270.
Based on the information from the previous problem, how many bats can Luis purchase for the same amount he would pay for three gloves?
From the previous problem, we know that g + 3b = 135 and 2g + b = 135, so we can set the two expressions equal to one another to get the equation g + 3b = 2g + b. Simplifying, we get 2b = g. That means each glove Luis purchases costs the same amount as two bats. Therefore, for the same amount paid for three gloves, Luis could purchase 3 × 2 = 6 bats.
Luis has $300 budgeted for the purchase of new baseball bats and gloves this spring. Luis needs to purchase a minimum of two new bats and two new gloves, and he wants to spend as much as possible of the budgeted amount. Based on the previous problems, what is the greatest number of bats Luis can purchase?
In the previous problem, we determined that 2b = g. We can substitute in the equation g + 3b = 135 to get 2b + 3b = 135. Simplifying and solving, we find that the price of a bat is 5b = 135 → b = $27. That means price of a glove is 2 × 27 = $54. Let’s first determine the greatest number of bats Luis can purchase along with two gloves. The price of two gloves is 2 × 54 = $108. That leaves 300 – 108 = $192 to spend on bats. Since 192 ÷ 27 = 7 1/9, Luis can purchase seven bats for a total of 27 × 7 = $189. Luis will spend 108 + 189 = $297 to purchase two gloves and seven bats. Now, let’s see if this maximizes the amount he can spend of the $300 budget. The GCF of 27 and 54 is 27, and 300 ÷ 27 = 11 1/9. Of the $300, Luis can spend no more than 27 × 11 = $297 purchasing bats and gloves. Therefore, using as much as possible of the budget, the greatest number of bats Luis can purchase (along with at least two gloves) is 7 bats.
♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦
Next Monday, April 15, is Tax Day!
It is estimated that the total time Americans will spend on taxes this year is 6.5 billion hours! According to the White House budget office, tax work accounts for approximately 70% of the paperwork burden of the federal government. If 6.5 billion hours is 70% of the total time spent on federal government paperwork, how many hours are equivalent to 50% of the total time spent on federal government paperwork? Express your answer as a whole number of hours rounded to the nearest million.
We are told the 6.5 billion hours is 70% of x, and we need to find 50% of x. We can either take the time to determine the value of x, or we can skip that step altogether using the following proportion: 6.5/0.7 = y/0.5. If we set the cross-products equal, we have 0.7y = 0.5(6.5) or 0.7y = 3.25. Dividing both sides by 0.7 yields y = 4.6428571429. Remember that this is really 46,428,571,429 hours, or 46,429,000,000 hours, rounded to the nearest million.
Say that 1.1 billion of the total hours is spent by people filing the most basic tax return. If it is estimated that the average filer of this basic tax return spends approximately 13 hours, 48 minutes, how many filers are in this “most basic tax return” category? Express your answer to the nearest hundred thousand.
We need to figure out how many different filers each spend 13 hours, 48 minutes in order to reach the total of 1.1 billion hours. To do this, we can divide the total by the part. However, before doing this, we need to figure out how 13 hours, 48 minutes can be represented in just hours. There are 60 minutes in an hour, so 48 minutes is 48 ÷ 60 = 0.8 hours. This means 13 hours, 48 minutes is equivalent to 13.8 hours. Now, we can calculate that there are 1,100,000,000 ÷ 13.8 = 79,710,144.927536 or 79,700,000 filers in this category, to the nearest hundred thousand.
In order to reduce the burden of the time spent on taxes, the IRS created the Office of Taxpayer Burden Reduction in early 2002. In its first five years, it shaved a total of about 200 million hours from tax paperwork. It would make sense that the office’s first year in existence may have led to the most improvement. Let’s assume that in its first year, it shaved 80 million hours, but the remainder of the 200 million hours was spread equally over the next four years and will continue at this same rate into the future. How many total hours would we have expected to be shaved by the end of its eighth year in existence? If y represents the total number of shaved hours and x is the total number of years the office has been in existence, what equation in the form y = mx + b represents this situation for x ³ 1?
If the first year accounted for 80 million hours, then there were still 200 – 80 = 120 million hours left. We are told that these hours were equally spread over the next four years, which is 120 ÷ 4 = 30 million each year. At the end of these five years, the total was 200 million. If we give the office three more years, that would be an additional 3(30) = 90 million hours, for a total of 290 million hours at the end of eight years. In order to come up with an equation that represents this situation, it might be easiest to look at some data points. (For this, we won’t use millions, but just the number of millions of hours.) We know that the points (1, 80), (2, 110), (3, 140), (4, 170), and (5, 200) should be generated from our equation, as well as (8, 290). After the first year, the y-value increases by 30 every time the x-value increases by 1. We also know, though, that we start off with 80 in the first year. So, how do we account for this? Something like y = 30x + 80 looks like it might work, since it accounts for the initial 80 and every time we increase x by 1, we’ll increase our total by 30. But check what happens if x = 1. We get y = 30(1) + 80 = 110, and that’s 30 too much. If we’re going to have to include one 30 during the first year, then we’re going to have to take 30 out of the 80. Let’s try y = 30x + 50. Though we never had the number 50 mentioned anywhere in the problem, this helps our situation when x = 1, and every time we increase x by 1, our y-value will go up by 30 as we need. This is it: y = 30x + 50.
♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦
The 3rd Thursday in April (April 18, 2024) is National High Five Day!
For National High Five Day, Ronnie’s class decides that everyone in the class should exchange one high five with each other person in the class. If there are 20 people in Ronnie’s class, how many high fives will be exchanged?
Each of the 20 people in Ronnie’s class will high five 19 people (everyone but themselves). However, if we simply multiply 20 × 19, we will have double-counted each of the high fives. For example, we will have counted when Ronnie high fives Ruth and when Ruth high fives Ronnie. This is actually the same high five, though, since each person in the class is exchanging only one high five with each other person. Thus, we’ll need to account for this by dividing by 2, so there will have been (20 × 19)/2 = 190 high fives.
At the end of Ronnie’s lacrosse game, every player on his team high-fived every member of the opposing team once and said “good game.” If each team had 15 members, how many high fives were exchanged?
If we look at team member #1 on Ronnie’s lacrosse team, we know that he will give 15 high fives, one to each of the 15 members on the opposite team. Team member #2 on Ronnie’s team will do the same, as will team member #3, and so on. Therefore, multiplying Ronnie’s 15 team members by the 15 team members on the opposite team gives 15 × 15 = 225 high fives exchanged.
In an effort to spread “high-fiving cheer” Ronnie and Ruth decided they would each high five 2 people and ask each person they high five to high five 2 additional people (and ask each of them to do the same). After 4 cycles like this, how many high fives will be exchanged?
Let’s track the high fives generated by Ronnie.
Ronnie high fives 2 people → 2 high fives
Those 2 people high five 2 more people each → 2(2) = 4 high fives
Those 4 people each high five 2 more people → 4(2) = 8 high fives
Those 8 people each high five 2 more people → 8(2) = 16 high fives
That’s a total of 2 + 4 + 8 + 16 = 30 high fives. Since Ruth did the same as Ronnie, together they would have generated 2(30) = 60 high fives.
♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦
In the spring of 2020, Dahlia created a flower bed and planted one daffodil. The following spring, three daffodils bloomed. In fact, each spring, from 2020 to 2023, the number of daffodils that bloomed in Dahlia’s flower bed was 1, 3, 9 and 27, respectively. If this pattern continues, how many daffodils can Dahlia expect to bloom in the spring of 2024?
It appears that each year after 2020, the number of daffodils that bloomed in Dahlia’s garden was three times the number that bloomed the previous year. If this pattern continues, in 2024, Dahlia can expect there to be 27 × 3 = 81 blooms.
In the spring of 2020, Rose created a flower bed and planted two tulips. The following spring four tulips bloomed. Each spring, from 2020 to 2023, the number of tulips that bloomed in Rose’s flower bed was 2, 4, 8 and 16, respectively. If this pattern continues, what is the absolute difference in the number of tulips and daffodils expected to bloom in Rose’s and Dahlia’s flower beds in 2024?
It appears that every year since Rose planted two tulips in 2020, the number of tulips that bloomed was twice the number that bloomed the previous year. If this pattern continues, Rose can expect 16 × 2 = 32 tulips to bloom in 2024. The difference in the number of tulips and daffodils that are expected to bloom in Rose’s and Dahlia’s gardens in 2024 is 81 – 32 = 49.
In the spring of 2020, Lily created a flower bed and planted three hyacinths. Each spring, from 2020 to 2023, the number of hyacinths that bloomed in Lily’s flower bed was 3, 6, 10 and 15, respectively. If this pattern continues, what is the total number of hyacinths that will have bloomed in Lily’s flower bed from the spring of 2020 through the spring of 2024?
Lily planted 3 hyacinths (which bloomed) in 2020. In 2021, the number of hyacinths that bloomed was three more than the number that bloomed in 2020. In 2022, the number of blooms in Lily’s garden was four more than the number of blooms in 2021, and in 2023, there were five more blooms than in 2022. If this pattern continues, Lily can expect the number of blooms in 2024 to be six more than in 2023, which would be 15 + 6 = 21 blooms. So, the total number of hyacinths that will have bloomed in Lily’s garden from 2020 through 2024 was 3 + 6 + 10 + 15 + 21 = 55 hyacinths.
♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦
According to the North American Numbering Plan Administration (NANPA) standards, U.S. phone numbers contain a 3-digit area code, followed by a 3-digit exchange code and end in a 4-digit subscriber number. The first digit of both the area code and exchange code cannot be 0 or 1. How many different combinations of area code and exchange code are possible? Express your answer in scientific notation.
In the state of Maryland, the 301 area code took effect in 1947, and by 1991 every possible phone number with a 301 area code had been assigned to a Maryland phone subscriber. At that time, the state instituted the 410 area code. Just six years later the state had exhausted its pool of available phone numbers with a 410 area code. So in 1997, the 240 and 443 area codes took effect in the state. In 2012, once again on the verge of running out of phone numbers, the state of Maryland introduced a new 667 area code to expand its pool of available phone numbers. What is the total number of unique phone numbers possible (in compliance with NANPA standards) using the 301, 410, 240, 443 and 667 area codes?
It took 44 years (1947 to 1991) to deplete the pool of available numbers with a 301 area code in the state of Maryland, but only 6 years (1991 to 1997) to exhaust all possible numbers with a 410 area code. The proliferation of cell phone use is the main cause for the drastic increase in the rate at which phone numbers were assigned. What is the percent increase from the rate at which 301 area code phone numbers were assigned to the rate at which numbers with a 410 area code were assigned? Express your answer to the nearest whole number.
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Luis plans to purchase some new baseball bats and gloves before spring training begins. Luis wants to buy the same bats and gloves he purchased last spring, but he can’t remember the price of each item. Luis recalls making two purchases last spring, each totaling $135 and both tax exempt. The first purchase was for a glove and three bats. The second purchase was for two gloves and a bat. If the current prices of these items are the same as last spring, how much will Luis pay this spring for three gloves and four bats?
If we let g and b represent the prices of a glove and a bat, respectively, we can derive two equations: g + 3b = 135 and 2g + b = 135. We want to determine the price of three gloves and four bats. Combined, the two purchases from last spring were for (g + 3b) + (2g + b) = 3g + 4b, which is three gloves and four bats. So, we can add the two totals to find that the price is 135 + 135 = $270.
Based on the information from the previous problem, how many bats can Luis purchase for the same amount he would pay for three gloves?
From the previous problem, we know that g + 3b = 135 and 2g + b = 135, so we can set the two expressions equal to one another to get the equation g + 3b = 2g + b. Simplifying, we get 2b = g. That means each glove Luis purchases costs the same amount as two bats. Therefore, for the same amount paid for three gloves, Luis could purchase 3 × 2 = 6 bats.
Luis has $300 budgeted for the purchase of new baseball bats and gloves this spring. Luis needs to purchase a minimum of two new bats and two new gloves, and he wants to spend as much as possible of the budgeted amount. Based on the previous problems, what is the greatest number of bats Luis can purchase?
In the previous problem, we determined that 2b = g. We can substitute in the equation g + 3b = 135 to get 2b + 3b = 135. Simplifying and solving, we find that the price of a bat is 5b = 135 → b = $27. That means price of a glove is 2 × 27 = $54. Let’s first determine the greatest number of bats Luis can purchase along with two gloves. The price of two gloves is 2 × 54 = $108. That leaves 300 – 108 = $192 to spend on bats. Since 192 ÷ 27 = 7 1/9, Luis can purchase seven bats for a total of 27 × 7 = $189. Luis will spend 108 + 189 = $297 to purchase two gloves and seven bats. Now, let’s see if this maximizes the amount he can spend of the $300 budget. The GCF of 27 and 54 is 27, and 300 ÷ 27 = 11 1/9. Of the $300, Luis can spend no more than 27 × 11 = $297 purchasing bats and gloves. Therefore, using as much as possible of the budget, the greatest number of bats Luis can purchase (along with at least two gloves) is 7 bats.
♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦
Next Monday, April 15, is Tax Day!
It is estimated that the total time Americans will spend on taxes this year is 6.5 billion hours! According to the White House budget office, tax work accounts for approximately 70% of the paperwork burden of the federal government. If 6.5 billion hours is 70% of the total time spent on federal government paperwork, how many hours are equivalent to 50% of the total time spent on federal government paperwork? Express your answer as a whole number of hours rounded to the nearest million.
We are told the 6.5 billion hours is 70% of x, and we need to find 50% of x. We can either take the time to determine the value of x, or we can skip that step altogether using the following proportion: 6.5/0.7 = y/0.5. If we set the cross-products equal, we have 0.7y = 0.5(6.5) or 0.7y = 3.25. Dividing both sides by 0.7 yields y = 4.6428571429. Remember that this is really 46,428,571,429 hours, or 46,429,000,000 hours, rounded to the nearest million.
Say that 1.1 billion of the total hours is spent by people filing the most basic tax return. If it is estimated that the average filer of this basic tax return spends approximately 13 hours, 48 minutes, how many filers are in this “most basic tax return” category? Express your answer to the nearest hundred thousand.
We need to figure out how many different filers each spend 13 hours, 48 minutes in order to reach the total of 1.1 billion hours. To do this, we can divide the total by the part. However, before doing this, we need to figure out how 13 hours, 48 minutes can be represented in just hours. There are 60 minutes in an hour, so 48 minutes is 48 ÷ 60 = 0.8 hours. This means 13 hours, 48 minutes is equivalent to 13.8 hours. Now, we can calculate that there are 1,100,000,000 ÷ 13.8 = 79,710,144.927536 or 79,700,000 filers in this category, to the nearest hundred thousand.
In order to reduce the burden of the time spent on taxes, the IRS created the Office of Taxpayer Burden Reduction in early 2002. In its first five years, it shaved a total of about 200 million hours from tax paperwork. It would make sense that the office’s first year in existence may have led to the most improvement. Let’s assume that in its first year, it shaved 80 million hours, but the remainder of the 200 million hours was spread equally over the next four years and will continue at this same rate into the future. How many total hours would we have expected to be shaved by the end of its eighth year in existence? If y represents the total number of shaved hours and x is the total number of years the office has been in existence, what equation in the form y = mx + b represents this situation for x ³ 1?
If the first year accounted for 80 million hours, then there were still 200 – 80 = 120 million hours left. We are told that these hours were equally spread over the next four years, which is 120 ÷ 4 = 30 million each year. At the end of these five years, the total was 200 million. If we give the office three more years, that would be an additional 3(30) = 90 million hours, for a total of 290 million hours at the end of eight years. In order to come up with an equation that represents this situation, it might be easiest to look at some data points. (For this, we won’t use millions, but just the number of millions of hours.) We know that the points (1, 80), (2, 110), (3, 140), (4, 170), and (5, 200) should be generated from our equation, as well as (8, 290). After the first year, the y-value increases by 30 every time the x-value increases by 1. We also know, though, that we start off with 80 in the first year. So, how do we account for this? Something like y = 30x + 80 looks like it might work, since it accounts for the initial 80 and every time we increase x by 1, we’ll increase our total by 30. But check what happens if x = 1. We get y = 30(1) + 80 = 110, and that’s 30 too much. If we’re going to have to include one 30 during the first year, then we’re going to have to take 30 out of the 80. Let’s try y = 30x + 50. Though we never had the number 50 mentioned anywhere in the problem, this helps our situation when x = 1, and every time we increase x by 1, our y-value will go up by 30 as we need. This is it: y = 30x + 50.
♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦
The 3rd Thursday in April (April 18, 2024) is National High Five Day!
For National High Five Day, Ronnie’s class decides that everyone in the class should exchange one high five with each other person in the class. If there are 20 people in Ronnie’s class, how many high fives will be exchanged?
Each of the 20 people in Ronnie’s class will high five 19 people (everyone but themselves). However, if we simply multiply 20 × 19, we will have double-counted each of the high fives. For example, we will have counted when Ronnie high fives Ruth and when Ruth high fives Ronnie. This is actually the same high five, though, since each person in the class is exchanging only one high five with each other person. Thus, we’ll need to account for this by dividing by 2, so there will have been (20 × 19)/2 = 190 high fives.
At the end of Ronnie’s lacrosse game, every player on his team high-fived every member of the opposing team once and said “good game.” If each team had 15 members, how many high fives were exchanged?
If we look at team member #1 on Ronnie’s lacrosse team, we know that he will give 15 high fives, one to each of the 15 members on the opposite team. Team member #2 on Ronnie’s team will do the same, as will team member #3, and so on. Therefore, multiplying Ronnie’s 15 team members by the 15 team members on the opposite team gives 15 × 15 = 225 high fives exchanged.
In an effort to spread “high-fiving cheer” Ronnie and Ruth decided they would each high five 2 people and ask each person they high five to high five 2 additional people (and ask each of them to do the same). After 4 cycles like this, how many high fives will be exchanged?
Let’s track the high fives generated by Ronnie.
Ronnie high fives 2 people → 2 high fives
Those 2 people high five 2 more people each → 2(2) = 4 high fives
Those 4 people each high five 2 more people → 4(2) = 8 high fives
Those 8 people each high five 2 more people → 8(2) = 16 high fives
That’s a total of 2 + 4 + 8 + 16 = 30 high fives. Since Ruth did the same as Ronnie, together they would have generated 2(30) = 60 high fives.
♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦
In the spring of 2020, Dahlia created a flower bed and planted one daffodil. The following spring, three daffodils bloomed. In fact, each spring, from 2020 to 2023, the number of daffodils that bloomed in Dahlia’s flower bed was 1, 3, 9 and 27, respectively. If this pattern continues, how many daffodils can Dahlia expect to bloom in the spring of 2024?
It appears that each year after 2020, the number of daffodils that bloomed in Dahlia’s garden was three times the number that bloomed the previous year. If this pattern continues, in 2024, Dahlia can expect there to be 27 × 3 = 81 blooms.
In the spring of 2020, Rose created a flower bed and planted two tulips. The following spring four tulips bloomed. Each spring, from 2020 to 2023, the number of tulips that bloomed in Rose’s flower bed was 2, 4, 8 and 16, respectively. If this pattern continues, what is the absolute difference in the number of tulips and daffodils expected to bloom in Rose’s and Dahlia’s flower beds in 2024?
It appears that every year since Rose planted two tulips in 2020, the number of tulips that bloomed was twice the number that bloomed the previous year. If this pattern continues, Rose can expect 16 × 2 = 32 tulips to bloom in 2024. The difference in the number of tulips and daffodils that are expected to bloom in Rose’s and Dahlia’s gardens in 2024 is 81 – 32 = 49.
In the spring of 2020, Lily created a flower bed and planted three hyacinths. Each spring, from 2020 to 2023, the number of hyacinths that bloomed in Lily’s flower bed was 3, 6, 10 and 15, respectively. If this pattern continues, what is the total number of hyacinths that will have bloomed in Lily’s flower bed from the spring of 2020 through the spring of 2024?
Lily planted 3 hyacinths (which bloomed) in 2020. In 2021, the number of hyacinths that bloomed was three more than the number that bloomed in 2020. In 2022, the number of blooms in Lily’s garden was four more than the number of blooms in 2021, and in 2023, there were five more blooms than in 2022. If this pattern continues, Lily can expect the number of blooms in 2024 to be six more than in 2023, which would be 15 + 6 = 21 blooms. So, the total number of hyacinths that will have bloomed in Lily’s garden from 2020 through 2024 was 3 + 6 + 10 + 15 + 21 = 55 hyacinths.
♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦
According to the North American Numbering Plan Administration (NANPA) standards, U.S. phone numbers contain a 3-digit area code, followed by a 3-digit exchange code and end in a 4-digit subscriber number. The first digit of both the area code and exchange code cannot be 0 or 1. How many different combinations of area code and exchange code are possible? Express your answer in scientific notation.
In the state of Maryland, the 301 area code took effect in 1947, and by 1991 every possible phone number with a 301 area code had been assigned to a Maryland phone subscriber. At that time, the state instituted the 410 area code. Just six years later the state had exhausted its pool of available phone numbers with a 410 area code. So in 1997, the 240 and 443 area codes took effect in the state. In 2012, once again on the verge of running out of phone numbers, the state of Maryland introduced a new 667 area code to expand its pool of available phone numbers. What is the total number of unique phone numbers possible (in compliance with NANPA standards) using the 301, 410, 240, 443 and 667 area codes?
It took 44 years (1947 to 1991) to deplete the pool of available numbers with a 301 area code in the state of Maryland, but only 6 years (1991 to 1997) to exhaust all possible numbers with a 410 area code. The proliferation of cell phone use is the main cause for the drastic increase in the rate at which phone numbers were assigned. What is the percent increase from the rate at which 301 area code phone numbers were assigned to the rate at which numbers with a 410 area code were assigned? Express your answer to the nearest whole number.
CHECK THE PROBLEM OF THE WEEK ARCHIVE
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