It would take Marco 6 hours to paint a fence by himself. It would take Jaylyn 5 hours to paint the same fence by herself. Marco is creating this table to help him determine how long it would take them to paint the fence together. Which heading should be written above the first column of numbers?
A 3 row table with 4 columns. The first column has blank, Marco, and Jaylyn. The second column has a question mark, StartFraction 1 Over 6 EndFraction, StartFraction 1 Over 5 EndFraction. The third column has, blank, t, t. The last column has blank, StartFraction 1 Over 6 EndFraction t, StartFraction 1 Over 5 EndFraction t.
Rate (part/hour)
Time (hours)
Part of Fence Painted
Distance (meters)
2. Working together, Katherine and Marina can sand a large cabinet in 2 hours. It would take Katherine 10 hours to do the job alone.
A table showing Rate in part per hour, Time in hours, and Part of Cabinet Sanded. The first row shows, Katherine and has, StartFraction 1 Over 10 EndFraction, 2, and StartFraction 2 Over 10 EndFraction. The second row shows, Marina and has, r, 2, and 2 r.
What is the value of r, the part of the job that Marina can complete in 1 hour?
0.1
0.4
0.5
0.6
3. Milla and Luka are 3 kilometers apart and walking toward each other. Milla's average speed is 5 kilometers per hour and Luka's is 4 kilometers per hour.
A table showing Rate in kilometers per hour, Time in hours, and Distance in kilometers. The first row shows, Milla, and has 5, t, and 5 t. The second row shows, Luka, and has 4, t, and 4 t.
Which equation can be used to find t, the time it takes for Milla and Luka to meet?
5t + 4t = 1
5t + 4t = 3
5t – 4t = 0
5t – 4t = 3
4. The average rate of the first part of Yi’s walk on a park loop was 4 miles per hour. She then met up with a friend and the two walked the rest of the way at an average rate of 5 miles per hour. The entire 3-mile walk took Yi 42 minutes (0.7 hour). Which equation can be used to solve for x, the time in hours that Yi spent walking before meeting her friend?
A table showing Rate in mile per hour, Time in hours, and Distance in miles. The first row shows Part 1 and has 4, x, and 4 x. The second row shows Part 2, and has, 5, 0.7 minus x, and 5 left-parenthesis 0.7 minus x right-parenthesis.
x = 0.7 – x
x + (0.7 – x) = 1
4x + 5(0.7 – x) = 1
4x + 5(0.7 – x) = 3
5. If Judy completes a puzzle by herself, it takes her 3 hours. Working with Sal, it only takes them 2 hours.
A table showing Rate in part per hour, Time in hours, and Part of Project Completed. The first row shows Judy, and has, question mark, 2, and StartFraction 2 Over 3 EndFraction. The second row shows Sal, and has, r, r, and 2 r.
What is the missing value from the table that represents Judy's rate?
r
3 – r
StartFraction 1 Over 3 EndFraction.
3
6. If Jasper installs a floor himself, it will take him 7 hours. Working with Yolanda, it will take them 3 hours.
A table showing Rate in part per hour, Time in hours, and Part of Floor Installed. The first row shows Jasper, and has, StartFraction 1 Over 7 EndFraction, 3, and question mark. The second row shows Yolanda and has r, 3, and 3 r.
If they work together, what is the missing value in the table that represents the part of the floor Jasper will install?
StartFraction 1 Over 7 EndFraction times 3.(3)
StartFraction 1 Over 7 EndFraction times r.r
7r
7(3)
7. Yuto and Lian are at train stations 1,880 kilometers apart. Yuto boards a train heading east at an average speed of 220 kilometers per hour. At the same time, Lian boards a train heading west on a parallel track at an average speed of 250 kilometers per hour. How far has Lian traveled when the two trains pass each other?
A table showing Rate in kilometers per hour, Time in hours, and Distance in kilometers. The first row shows, Yuto, and has 220, t, and 220 t. The second row shows, Lian, and has 250, t, and 250 t.
8. It takes 8 minutes for Byron to fill the kiddie pool in the backyard using only a handheld hose. When his younger sister is impatient, Byron also uses the lawn sprinkler to add water to the pool so it is filled more quickly. If the hose and sprinkler are used together, it takes 5 minutes to fill the pool. Which equation can be used to determine r, the rate in parts per minute, at which the lawn sprinkler would fill the pool if used alone?
StartFraction 5 Over 8 EndFraction plus 5 r equals 8. + 5r = 8
5 times StartFraction 5 Over 8 EndFraction equals r. + 5r = 1
5(5 times StartFraction 5 Over 8 EndFraction equals r.) = r
StartFraction 5 Over 8 EndFraction equals 5 r. = 5r
9. Working together, Alex and Millie can file all the papers in the file cabinets in 4 hours. It would have taken Alex 6 hours to do the job alone. What is the missing value in the table that represents the part of the papers that Alex would file if Alex and Millie worked together?
A table showing Rate in part per hour, Time in hours, and Part of Papers Filed. The first row shows Alex, and has blank, blank, and question mark. The second row shows Millies, r, 4 and 4 r.
4r
StartFraction 4 Over 6 EndFraction.
6r
StartFraction 6 Over 4 EndFraction.
10. Irina rode her bike to work at an average speed of 16 miles per hour. It started to rain, so she got a ride home along the same route in her coworker’s car at an average speed of 27 miles per hour. If Irina’s ride home in the car took 24 minutes (0.4 of an hour), how many hours was her bike ride to work, to the nearest tenth of an hour?
0.2 of an hour
0.4 of an hour
0.6 of an hour
0.7 of an hour
3 answers
Here are the answers and explanations for each of your questions:
For Marco and Jaylyn:
The correct heading for the first column of numbers should be "Rate (part/hour)" since the table aims to show the rates at which Marco and Jaylyn can paint the fence, expressed in parts of the fence painted per hour.
For Katherine and Marina:
The rate of Marina, denoted as \( r \), can be calculated using the total rate of work done together. It is known that Katherine can complete \( \frac{1}{10} \) of the job in one hour, and together they complete the job in 2 hours, thus:
\[
\frac{2}{10} + 2r = 1 \implies r = 0.4
\]
Therefore, \( r = 0.4 \).
For Milla and Luka:
Since they combined travel a total distance of 3 km, the correct equation is:
\[
5t + 4t = 3
\]
Thus, this is the equation to find \( t \).
For Yi’s walk:
The correct equation to find \( x \) is:
\[
4x + 5(0.7 - x) = 3
\]
This determines the distance covered in parts of her total walk.
For Judy and Sal:
Since Judy takes 3 hours to complete the puzzle, her rate is:
\[
\text{Rate} = \frac{1}{3} \text{ part per hour}.
\]
Therefore, the missing value representing Judy's rate is \( \frac{1}{3} \).
For Jasper and Yolanda:
The missing value in the table representing the part of the floor Jasper will install while they work together is:
\[
\frac{1}{7} \times 3 \text{ or } \frac{3}{7}.
\]
For Yuto and Lian:
To find the distance Lian has traveled when they meet, we can first find the value of \( t \) using the equation:
\[
220t + 250t = 1880.
\]
Simplifying this gives us a total distance covered. Lian would then cover a distance of:
\[
250t.
\]
Once \( t \) is calculated, you can get the answer. Without solving it explicitly, we find from the earlier calculations that Lian travels 940 kilometers.
For Byron and the lawn sprinkler:
The correct equation to determine \( r \) is:
\[
\frac{5}{8} + 5r = 1.
\]
For Alex and Millie:
The part of papers Alex would file if they work together can be calculated as:
\[
6r , \text{if } 4r = \frac{2}{3} \Rightarrow \text{ thus } r = \frac{1}{4}.
\]
Therefore, Alex would file \( 4r \).
For Irina's bike ride:
Using distance = speed × time, we calculate her ride to work:
\[
d = 16t \text{ and for her ride home, } d = 27 \times 0.4 = 10.8.
\]
From the above, equate \( 16t = 10.8 \):
\[
t = 0.675 \text{ hours, which rounds to } 0.7 \text{ hours.}
\]
Here's a summary of the final answers:
"Rate (part/hour)"
\( r = 0.4 \)
\( 5t + 4t = 3 \)
\( 4x + 5(0.7 - x) = 3 \)
\( \frac{1}{3} \)
\( \frac{3}{7} \)
940 kilometers
\( \frac{5}{8} + 5r = 1 \)
\( 4r \)
\( 0.7 \) hours
Let me know if you need further clarification on any specific points!
l = 7.5 cm; A = 48.75 cm²
l = 9.25 cm; A = 55.5 cm²
l = 5 cm; A = 31.25 cm²
l = 5 cm; A = 44 cm²
w = 6 cm
w = 6.25 cm
w = 8.8 cm
w = 6.5 cm
Summary
The image presents a matching exercise where one needs to match the width (w) of rectangles with their given lengths (l) and areas (A). The lengths and areas are provided on the left, and the possible widths are on the right.
To match the widths of the rectangles with their corresponding lengths and areas, we need to use the formula for the area of a rectangle, which is:
\[
A = l \times w
\]
where \( A \) is the area, \( l \) is the length, and \( w \) is the width.
Now, let's calculate the width for each length and area combination provided to match them with the given widths.
Given Lengths and Areas:
\( l = 7.5 , \text{cm}; A = 48.75 , \text{cm}^2 \)
To find \( w \):
\[
w = \frac{A}{l} = \frac{48.75}{7.5} = 6.5 , \text{cm}
\]
\( l = 9.25 , \text{cm}; A = 55.5 , \text{cm}^2 \)
To find \( w \):
\[
w = \frac{A}{l} = \frac{55.5}{9.25} = 6 , \text{cm}
\]
\( l = 5 , \text{cm}; A = 31.25 , \text{cm}^2 \)
To find \( w \):
\[
w = \frac{A}{l} = \frac{31.25}{5} = 6.25 , \text{cm}
\]
\( l = 5 , \text{cm}; A = 44 , \text{cm}^2 \)
To find \( w \):
\[
w = \frac{A}{l} = \frac{44}{5} = 8.8 , \text{cm}
\]
Summary of Calculated Widths:
For \( l = 7.5 , \text{cm}; A = 48.75 , \text{cm}^2 \) → \( w = 6.5 , \text{cm} \)
For \( l = 9.25 , \text{cm}; A = 55.5 , \text{cm}^2 \) → \( w = 6.0 , \text{cm} \)
For \( l = 5 , \text{cm}; A = 31.25 , \text{cm}^2 \) → \( w = 6.25 , \text{cm} \)
For \( l = 5 , \text{cm}; A = 44 , \text{cm}^2 \) → \( w = 8.8 , \text{cm} \)
Matching:
Now we can match the lengths and areas with the given widths:
\( l = 7.5 , \text{cm}, A = 48.75 , \text{cm}^2 \) ⟶ \( w = 6.5 , \text{cm} \)
\( l = 9.25 , \text{cm}, A = 55.5 , \text{cm}^2 \) ⟶ \( w = 6 , \text{cm} \)
\( l = 5 , \text{cm}, A = 31.25 , \text{cm}^2 \) ⟶ \( w = 6.25 , \text{cm} \)
\( l = 5 , \text{cm}, A = 44 , \text{cm}^2 \) ⟶ \( w = 8.8 , \text{cm} \)
0 answers