Patterns, Relationships, and Algebraic Reasoning Quiz

1) The state of Ohio has 88 counties. Texas has 10 fewer than three times as many counties. How many counties are in Texas?

264

254

244

39

To find out how many counties are in Texas, we need to follow these steps:

Determine the number of counties in Ohio: Ohio has 88 counties.

Calculate three times the number of counties in Ohio:

3×88=264

Subtract 10 from this amount to find the number of counties in Texas:

264−10=254

So, Texas has 254 counties.

Explanation

To find out how many counties are in Texas, we need to follow these steps:

Determine the number of counties in Ohio: Ohio has 88 counties.

Calculate three times the number of counties in Ohio:

3×88=264

Subtract 10 from this amount to find the number of counties in Texas:

264−10=254

So, Texas has 254 counties.

2)

18

27

39

54

The image provides us with a handy formula to calculate the number of diagonals in a polygon:

Number of diagonals = n(n-3) / 2

where 'n' represents the number of sides of the polygon.

Applying the Formula

We are asked to find the number of diagonals in a polygon with 9 sides.

Substitute 'n' with 9 in the formula:

Number of diagonals = 9(9 - 3) / 2

Simplify the equation:

Number of diagonals = 9(6) / 2 Number of diagonals = 54 / 2 Number of diagonals = 27

Explanation

The image provides us with a handy formula to calculate the number of diagonals in a polygon:

Number of diagonals = n(n-3) / 2

where 'n' represents the number of sides of the polygon.

Applying the Formula

We are asked to find the number of diagonals in a polygon with 9 sides.

Substitute 'n' with 9 in the formula:

Number of diagonals = 9(9 - 3) / 2

Simplify the equation:

Number of diagonals = 9(6) / 2 Number of diagonals = 54 / 2 Number of diagonals = 27

3)

15 miles

20 miles

25 miles

30 miles

1. Convert the mixed number to a fraction:

2 2/5 hours = (2*5 + 2)/5 = 12/5 hours

2. Find the duck's speed:

Speed = Distance / Time

Speed = 12 miles / (12/5 hours)

Speed = 12 miles * (5/12)

Speed = 5 miles per hour

3. Calculate the distance traveled in 5 hours:

Distance = Speed * Time

Distance = 5 miles/hour * 5 hours

Distance = 25 miles

Answer: The duck will travel about 25 miles in 5 hours.

Explanation

1. Convert the mixed number to a fraction:

2 2/5 hours = (2*5 + 2)/5 = 12/5 hours

2. Find the duck's speed:

Speed = Distance / Time

Speed = 12 miles / (12/5 hours)

Speed = 12 miles * (5/12)

Speed = 5 miles per hour

3. Calculate the distance traveled in 5 hours:

Distance = Speed * Time

Distance = 5 miles/hour * 5 hours

Distance = 25 miles

Answer: The duck will travel about 25 miles in 5 hours.

4) Lorenzo dropped a ball from an open window 50 feet above the ground in the Tower of the Americas in San Antonio, Texas. If the ball rebounds half the distance it drops each time it bounces, what will be the height of the rebound after the third bounce?

3.125 ft

6.25 ft

12.5 ft

25 ft

To find the height of the rebound after the third bounce, here’s the step-by-step breakdown:

Initial drop height: 50 feet.

After each bounce, the ball rebounds to half of the height it fell from.

First bounce: After dropping 50 feet, the ball rebounds to half of 50 feet.

Rebound height = 50 ÷ 2 = 25 feet.

Second bounce: After the ball rebounds 25 feet, it bounces back to half of 25 feet.

Rebound height = 25 ÷ 2 = 12.5 feet.

Third bounce: After rebounding 12.5 feet, it bounces back to half of 12.5 feet.

Rebound height = 12.5 ÷ 2 = 6.25 feet.

Answer: The height of the ball’s rebound after the third bounce is 6.25 feet.

Explanation

To find the height of the rebound after the third bounce, here’s the step-by-step breakdown:

Initial drop height: 50 feet.

After each bounce, the ball rebounds to half of the height it fell from.

First bounce: After dropping 50 feet, the ball rebounds to half of 50 feet.

Rebound height = 50 ÷ 2 = 25 feet.

Second bounce: After the ball rebounds 25 feet, it bounces back to half of 25 feet.

Rebound height = 25 ÷ 2 = 12.5 feet.

Third bounce: After rebounding 12.5 feet, it bounces back to half of 12.5 feet.

Rebound height = 12.5 ÷ 2 = 6.25 feet.

Answer: The height of the ball’s rebound after the third bounce is 6.25 feet.

5) Ms. Jamison spends $75 each month eating out at fast food restaurants. She plans to reduce her spending for fast food by $5 each month until she has reduced her spending to $25 per month. Which equation can be used to determine m, the number of months it will take for Ms. Jamison to reduce her fast food spending to $25 per month?

75-5m=25

75+5m=25

25-5m=75

25+5m=75

Initial Spending: Ms. Jamison starts with $75 per month.

Reduction Rate: She plans to reduce her spending by $5 each month, so after m months, her total reduction will be 5m.

Final Spending Goal: She wants to reduce her spending to $25 per month.

The equation to express this is:

Start with $75 and subtract 5m5m5m, where m is the number of months.

Set the equation equal to $25, as that’s the amount she wants to spend eventually.

Thus, the equation is:

75−5m=25

To solve for m, you would:

Subtract 25 from both sides: 75−5m=25 becomes 50=5m.

Divide both sides by 5: m=10.

It will take Ms. Jamison 10 months to reduce her fast food spending to $25 per month.

Explanation

Initial Spending: Ms. Jamison starts with $75 per month.

Reduction Rate: She plans to reduce her spending by $5 each month, so after m months, her total reduction will be 5m.

Final Spending Goal: She wants to reduce her spending to $25 per month.

The equation to express this is:

Start with $75 and subtract 5m5m5m, where m is the number of months.

Set the equation equal to $25, as that’s the amount she wants to spend eventually.

Thus, the equation is:

75−5m=25

To solve for m, you would:

Subtract 25 from both sides: 75−5m=25 becomes 50=5m.

Divide both sides by 5: m=10.

It will take Ms. Jamison 10 months to reduce her fast food spending to $25 per month.

6) Ashley mows lawns to earn money. She buys a new mower for $200 and charges $30 per lawn. If she mows n lawns, which of the following equations could you use to find p, Ashley's profit?

P=30n-100

P=30n-150

P=30n-200

P=30n-250

To find Ashley's profit p, you can use the following equation:

p = 30n - 200

Here’s how it’s derived:

Revenue: Ashley charges $30 per lawn. If she mows n lawns, her revenue is 30n dollars.

Expenses: She buys a new mower for $200. This is a one-time cost.

Profit Calculation: Profit is revenue minus expenses, so:

p=30n−200

Explanation

To find Ashley's profit p, you can use the following equation:

p = 30n - 200

Here’s how it’s derived:

Revenue: Ashley charges $30 per lawn. If she mows n lawns, her revenue is 30n dollars.

Expenses: She buys a new mower for $200. This is a one-time cost.

Profit Calculation: Profit is revenue minus expenses, so:

p=30n−200

7) Ivan drove 318 miles on the first day of his trip from Austin, Texas, to New Orleans, Louisiana, and 194 miles on the second day. His average speed was s miles per hour the first day and t miles per hour the second day. Which equation can be used to find p, the time it took Ivan to complete his trip?

P = 318 / s + 194 / t

P = s × 318 + t × 194

P = (318 + 194) / (s + t)

P = 318 / t + 194 / s

To find the total time p it took Ivan to complete his trip, you need to determine how long he spent driving each day and then add those times together.

Calculate Time for Each Day:

First Day: Ivan drove 318 miles at an average speed of s miles per hour. The time t1 spent driving on the first day is calculated by dividing the distance by the speed:

Time for the first day: t1 = 318 / s

Second Day: Ivan drove 194 miles at an average speed of t miles per hour. The time t2 spent driving on the second day is:

Time for the second day: t2 = 194 / t

Add the Times:

To get the total time p for the entire trip, sum the time for both days:

Total time: p = t1 + t2

Which is: p = 318 / s + 194 / t

Explanation

To find the total time p it took Ivan to complete his trip, you need to determine how long he spent driving each day and then add those times together.

Calculate Time for Each Day:

First Day: Ivan drove 318 miles at an average speed of s miles per hour. The time t1 spent driving on the first day is calculated by dividing the distance by the speed:

Time for the first day: t1 = 318 / s

Second Day: Ivan drove 194 miles at an average speed of t miles per hour. The time t2 spent driving on the second day is:

Time for the second day: t2 = 194 / t

Add the Times:

To get the total time p for the entire trip, sum the time for both days:

Total time: p = t1 + t2

Which is: p = 318 / s + 194 / t

8)

Wendy walked about twice as fast as Levina.

Wendy started her walk home later than Levina.

Wendy had been walking for about half as long as Levina when she overtook Levina.

Wendy had walked about half as far as Levina when she overtook Levina.

A. Wendy walked about twice as fast as Levina.

The slope of each line represents the speed of each person. Wendy's line is steeper, indicating she walked faster. It's difficult to say exactly "twice as fast" just from the graph, but it's definitely faster. So this statement could be true.

B. Wendy started her walk home later than Levina.

The graph shows Levina's line starting earlier on the time axis, meaning she left school first. This statement is true.

C. Wendy had been walking for about half as long as Levina when she overtook Levina.

The point where the lines intersect is when Wendy overtakes Levina. Looking at the time axis at this point, we see that Wendy had been walking for a shorter duration than Levina. This statement could be true.

D. Wendy had walked about half as far as Levina when she overtook Levina.

The point where the lines intersect is when Wendy overtakes Levina. At this point, they are the SAME distance from school. This statement is false.

Explanation

A. Wendy walked about twice as fast as Levina.

The slope of each line represents the speed of each person. Wendy's line is steeper, indicating she walked faster. It's difficult to say exactly "twice as fast" just from the graph, but it's definitely faster. So this statement could be true.

B. Wendy started her walk home later than Levina.

The graph shows Levina's line starting earlier on the time axis, meaning she left school first. This statement is true.

C. Wendy had been walking for about half as long as Levina when she overtook Levina.

The point where the lines intersect is when Wendy overtakes Levina. Looking at the time axis at this point, we see that Wendy had been walking for a shorter duration than Levina. This statement could be true.

D. Wendy had walked about half as far as Levina when she overtook Levina.

The point where the lines intersect is when Wendy overtakes Levina. At this point, they are the SAME distance from school. This statement is false.

9) Jane buys a sweater for $40 that was originally priced at $80. What percentage discount did Jane receive?

40%

50%

60%

70%

To find the percentage discount, use the formula:

Percentage Discount = (Original Price - Sale Price) ÷ Original Price × 100

Original Price = $80

Sale Price = $40

Discount = $80 - $40 = $40

Percentage Discount = ($40 ÷ $80) × 100 = 50%

Thus, Jane received a 50% discount on the sweater.

Explanation

To find the percentage discount, use the formula:

Percentage Discount = (Original Price - Sale Price) ÷ Original Price × 100

Original Price = $80

Sale Price = $40

Discount = $80 - $40 = $40

Percentage Discount = ($40 ÷ $80) × 100 = 50%

Thus, Jane received a 50% discount on the sweater.

10) A bookstore sells 3 notebooks for $6. If the store wants to increase the price by 20%, what will the new price for 3 notebooks be?

$7.00

$6.20

$7.20

$6.80

To calculate the price after a 20% increase, use the formula:

New Price = Original Price × (1 + Percentage Increase)

Original Price for 3 notebooks = $6

Percentage Increase = 20% = 0.20

New Price = $6 × (1 + 0.20)

New Price = $6 × 1.20 = $7.20

Therefore, the new price for 3 notebooks will be $7.20.

Explanation

To calculate the price after a 20% increase, use the formula:

New Price = Original Price × (1 + Percentage Increase)

Original Price for 3 notebooks = $6

Percentage Increase = 20% = 0.20

New Price = $6 × (1 + 0.20)

New Price = $6 × 1.20 = $7.20

Therefore, the new price for 3 notebooks will be $7.20.

Patterns, Relationships, And Algebraic Reasoning Quiz