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Line plots with fractional units

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5402763
Two board lengths are \(3\frac{1}{4}\) feet and \(3\frac{3}{4}\) feet. Where should the two Xs be placed on a line plot?

Hints

- Read each measured length exactly as written. - Match each measurement to the same value on the line-plot scale. - Use one X for each board length.

Solution

1. Each X must be placed above its exact measurement. 2. Place one X above \(3\frac{1}{4}\). 3. Place the other X above \(3\frac{3}{4}\).

Answer

Place one X at \(3\frac{1}{4}\) feet and one X at \(3\frac{3}{4}\) feet.
5402843
The line plot below shows measurements. Each X represents one measurement. How many more measurements are at \(2\frac{1}{4}\) inches than at \(2\frac{1}{2}\) inches?
Figure for problem 540284

Hints

- Each X represents one measurement. - Compare the numbers of Xs, not the distance between the labels. - Subtract the shorter stack from the taller stack.

Solution

1. Compare the two stack heights: \(5\) Xs and \(2\) Xs. 2. Subtract: \(5 - 2 = 3\).

Answer

There are \(3\) more measurements at \(2\frac{1}{4}\) inches.
5403353
The line plot below shows measurements. Each X represents one measurement. How many Xs must be removed from the taller stack so the two stacks have equal heights?
Figure for problem 540335

Hints

- Compare the two stack heights. - The taller stack must be reduced to match the shorter stack.

Solution

1. Find the difference in the stack heights: \(5 - 3 = 2\). 2. Removing \(2\) Xs from the taller stack leaves \(3\) Xs in each stack.

Answer

Remove \(2\) Xs from the stack at \(1\frac{1}{2}\) inches.
5403643
Six leaves were measured to the nearest quarter inch. Two leaves are \(1\frac{1}{4}\) inches long, two are \(1\frac{1}{2}\) inches long, and two are \(1\frac{3}{4}\) inches long. How many Xs should appear above each value on a line plot?

Hints

- Each measured leaf contributes one X. - Match each leaf length to the same value on the scale. - Count how many leaves have each length before placing the Xs.

Solution

1. Each X represents one measured leaf. 2. Place \(2\) Xs above \(1\frac{1}{4}\) inches. 3. Place \(2\) Xs above \(1\frac{1}{2}\) inches. 4. Place \(2\) Xs above \(1\frac{3}{4}\) inches.

Answer

Place \(2\) Xs at each of the three lengths: \(1\frac{1}{4}\), \(1\frac{1}{2}\), and \(1\frac{3}{4}\) inches.
5403693
The line plot below shows pencil lengths. Each X represents one pencil. One new pencil measurement is added at a length that already has at least one X. How many measurements and how many different measured lengths does the plot show now?
Figure for problem 540369

Hints

- One added pencil measurement adds one X to the plot. - Check whether the X creates a new scale position or joins an existing stack.

Solution

1. Adding one X changes the total number of measurements from \(8\) to \(9\). 2. The new X joins an existing stack, so the number of different measured values stays \(3\).

Answer

The line plot now shows \(9\) measurements at \(3\) different values.
5548573
The line plot shows ribbon lengths measured to the nearest quarter inch. Which length occurs most often?
Figure for problem 554857

Hints

- Each X represents one ribbon measurement. - Compare the heights of the stacks above the labeled quarter-inch values. - The most frequent length has the tallest stack.

Solution

1. Count the Xs above each value. 2. The stacks have heights \(1, 2, 3, 0\) from \(1\) inch through \(1\frac{3}{4}\) inches. 3. The tallest stack is above \(1\frac{1}{2}\) inches.

Answer

\(1\frac{1}{2}\) inches occurs most often.
5402363
Zara measured eight ribbon pieces to the nearest quarter inch. The lengths were \(3\), \(3\frac{1}{4}\), \(3\frac{1}{2}\), \(3\frac{1}{4}\), \(3\frac{3}{4}\), \(3\frac{1}{2}\), \(3\frac{1}{4}\), and \(4\) inches. How many Xs should appear above each value on a line plot?

Hints

- List the possible quarter-inch marks from the shortest length to the longest length. - Tally each measurement before deciding how many Xs belong above a mark.

Solution

1. Count how often each length appears. 2. \(3\) appears once; \(3\frac{1}{4}\) appears three times; \(3\frac{1}{2}\) appears twice; \(3\frac{3}{4}\) appears once; \(4\) appears once.

Answer

\(3\): \(1\) X \(3\frac{1}{4}\): \(3\) Xs \(3\frac{1}{2}\): \(2\) Xs \(3\frac{3}{4}\): \(1\) X \(4\): \(1\) X
5402463
The line plot below shows shell lengths. Each X represents one shell. Which lengths occur most often, and how many shells are shown altogether?
Figure for problem 540246

Hints

- Compare the stack heights above the three measured lengths. - A taller stack represents more shells of that length. - Add the Xs in all the stacks to find the total number of shells.

Solution

1. The tallest stacks each have \(4\) Xs. 2. The stacks at \(2\frac{1}{4}\) inches and \(2\frac{3}{4}\) inches each have \(4\) Xs. 3. Count all the shells: \(4 + 2 + 4 = 10\).

Answer

The lengths \(2\frac{1}{4}\) inches and \(2\frac{3}{4}\) inches have the most shells. The line plot shows \(10\) shells altogether.
5402523
A set of leaf lengths includes \(4\frac{1}{4}\) inches, \(4\frac{1}{2}\) inches, and \(4\frac{3}{4}\) inches. A student labels a line-plot scale only at \(4\), \(4\frac{1}{2}\), and \(5\). What two labels must be added so every measurement can be plotted exactly?

Hints

- Look at the smallest fractional step used by the measurements. - List the quarter-inch values in order from \(4\) to \(5\).

Solution

1. The data use quarter-inch measurements between \(4\) and \(5\). 2. The missing quarter-inch labels are \(4\frac{1}{4}\) and \(4\frac{3}{4}\).

Answer

Add \(4\frac{1}{4}\) and \(4\frac{3}{4}\).
5402563
A set of plant heights is measured in inches: \(6\), \(6\frac{1}{2}\), \(7\), and \(7\frac{1}{2}\). What is the simplest equal interval for the line-plot scale, and what labels should appear from the least value to the greatest value?

Hints

- Look for the smallest fractional step that appears in the data. - Make sure equal spaces on the scale represent equal numerical changes.

Solution

1. The measurements change by halves of an inch. 2. A half-inch interval shows every value exactly. 3. The labels are \(6\), \(6\frac{1}{2}\), \(7\), and \(7\frac{1}{2}\).

Answer

Use \(\frac{1}{2}\)-inch intervals. Label the scale \(6\), \(6\frac{1}{2}\), \(7\), and \(7\frac{1}{2}\).
5402723
A line plot has \(2\) Xs at \(2\frac{1}{2}\) inches, \(1\) X at \(2\frac{3}{4}\) inches, and \(3\) Xs at \(3\) inches. Write the complete data set in order from least to greatest.

Hints

- Each X stands for one measurement equal to the value below it. - Keep repeated measurements in the list rather than writing each value only once.

Solution

1. Repeat each measurement once for each X above it. 2. The ordered data are \(2\frac{1}{2}\), \(2\frac{1}{2}\), \(2\frac{3}{4}\), \(3\), \(3\), and \(3\) inches.

Answer

\(2\frac{1}{2}\), \(2\frac{1}{2}\), \(2\frac{3}{4}\), \(3\), \(3\), \(3\) inches
5402903
On a quarter-inch line-plot scale, the shortest measurement is \(2\frac{1}{4}\) inches and the longest is \(3\) inches. How many quarter-inch intervals separate the shortest and longest measurements?

Hints

- Move along the scale one quarter inch at a time. - Include both endpoints when listing values, but count the spaces between them. - Do not count the labeled values themselves as intervals.

Solution

1. List the scale values from shortest to longest: \(2\frac{1}{4}\), \(2\frac{1}{2}\), \(2\frac{3}{4}\), and \(3\). 2. Count the spaces between consecutive values: there are \(3\) quarter-inch intervals.

Answer

There are \(3\) quarter-inch intervals.
5402953
The line plot below shows measured lengths. Each X represents one measurement. One of the Xs above \(1\frac{1}{4}\) inches was recorded twice by mistake. After removing the duplicate, what are the two stack heights, how many measurements remain, and are the stacks tied?
Figure for problem 540295

Hints

- A duplicate is removed rather than moved to another value. - Recount the total and compare the two corrected stack heights.

Solution

1. Remove the duplicate X from \(1\frac{1}{4}\): \(3 - 1 = 2\) Xs. 2. The \(1\frac{1}{2}\)-inch stack remains at \(2\) Xs. 3. The corrected plot has \(2 + 2 = 4\) measurements, and the stack heights are equal.

Answer

Each stack has \(2\) Xs, \(4\) measurements remain, and the stacks are tied.
5403073
A line-plot scale is marked in quarter-inch intervals. How many scale values lie strictly between \(1\frac{1}{2}\) inches and \(2\frac{1}{2}\) inches? List them.

Hints

- Count forward by equal fourths. - “Strictly between” means neither endpoint is included.

Solution

1. Count by quarters after \(1\frac{1}{2}\): \(1\frac{3}{4}\), \(2\), \(2\frac{1}{4}\), then \(2\frac{1}{2}\). 2. Excluding both endpoints leaves \(3\) values.

Answer

There are \(3\) values: \(1\frac{3}{4}\), \(2\), and \(2\frac{1}{4}\) inches.
5403103
A quarter-inch line plot is labeled \(1\), \(1\frac{1}{4}\), \(1\frac{1}{2}\), \(1\frac{3}{4}\), and \(2\) inches. No measurement equals \(1\frac{1}{2}\) inches. A student wants to erase that label and close the gap. Should the label be removed? Explain.

Hints

- Separate the scale’s structure from the locations that happen to have measurements. - A labeled value does not need to have an X above it. - Consider what removing one value would do to equal spacing.

Solution

1. A line-plot scale must keep equal quarter-inch intervals across the entire range. 2. A scale value can have no Xs above it and still remain labeled. 3. Removing the label and closing the gap would make the spacing or sequence misleading.

Answer

No. The \(1\frac{1}{2}\)-inch label should remain so the scale keeps equal quarter-inch intervals.
5403253
The line plot below shows measurements. Each X represents one measurement. How many measurements are from \(1\frac{1}{4}\) inches through \(1\frac{3}{4}\) inches, including both endpoints?
Figure for problem 540325

Hints

- List every quarter-inch value from the first endpoint to the last. - Include both endpoints and the value between them. - Add the Xs at all included values.

Solution

1. Include the stacks at \(1\frac{1}{4}\), \(1\frac{1}{2}\), and \(1\frac{3}{4}\) inches. 2. Add the Xs: \(2 + 3 + 1 = 6\).

Answer

There are \(6\) measurements in the interval.
5403403
The data are \(2\), \(2\), \(2\frac{1}{4}\), \(2\frac{1}{4}\), \(2\frac{1}{2}\), \(2\frac{3}{4}\), \(2\frac{3}{4}\), and \(2\frac{3}{4}\) inches. A line plot shows \(2\) Xs at \(2\), \(2\) Xs at \(2\frac{1}{4}\), \(1\) X at \(2\frac{1}{2}\), and only \(2\) Xs at \(2\frac{3}{4}\). What is wrong, and how should it be corrected?

Hints

- Tally each repeated measurement in the data list. - Compare every tally with the corresponding line-plot stack.

Solution

1. The data contain \(3\) measurements of \(2\frac{3}{4}\) inches. 2. The plot shows only \(2\) Xs at that value. 3. Add one X above \(2\frac{3}{4}\) inches.

Answer

One X is missing at \(2\frac{3}{4}\) inches. The corrected stack should have \(3\) Xs.
5403463
A quarter-inch line-plot scale is labeled in this order: \(3\), \(3\frac{1}{4}\), \(3\frac{3}{4}\), \(3\frac{1}{2}\), \(4\). What is wrong? Write the labels in the correct order.

Hints

- Read the scale from least to greatest. - Check that every adjacent pair differs by one quarter inch.

Solution

1. Equal spaces must increase by \(\frac{1}{4}\) inch each time. 2. The labels \(3\frac{1}{2}\) and \(3\frac{3}{4}\) are reversed. 3. The correct order is \(3\), \(3\frac{1}{4}\), \(3\frac{1}{2}\), \(3\frac{3}{4}\), \(4\).

Answer

The middle two labels are out of order. The correct scale is \(3\), \(3\frac{1}{4}\), \(3\frac{1}{2}\), \(3\frac{3}{4}\), \(4\).
5403543
A line-plot scale is labeled \(2\), \(2\frac{1}{4}\), \(2\frac{1}{2}\), \(2\frac{3}{4}\), and \(3\), but the spaces between the labels are not equal. Is the scale valid? Explain what must be fixed.

Hints

- Compare the numerical step from one label to the next. - A graph scale must represent equal numerical changes with equal physical spacing.

Solution

1. Each numerical change is \(\frac{1}{4}\), so each physical interval must have the same length. 2. Unequal spacing would misrepresent the measurement distances. 3. The labels should be placed at equally spaced marks in the same order.

Answer

No. The scale is not valid until the quarter-inch labels are placed at equal intervals.
5403753
A quarter-inch line-plot scale runs from \(3\) inches to \(4\) inches. How many equal intervals are there, and how many scale labels are there when both endpoints are included?

Hints

- Intervals are the spaces between neighboring labels. - Including both endpoints makes the number of labels one more than the number of intervals.

Solution

1. The intervals are \(3\) to \(3\frac{1}{4}\), to \(3\frac{1}{2}\), to \(3\frac{3}{4}\), to \(4\). 2. There are \(4\) intervals and \(5\) labels.

Answer

There are \(4\) equal intervals and \(5\) scale labels.
5403843
The line plot below shows ribbon lengths. Each X represents one ribbon. One new ribbon measuring \(2\frac{1}{2}\) inches is added. How many different lengths are now shown, and what is the greatest number of Xs in any stack?
Figure for problem 540384

Hints

- Decide whether the new measurement uses a previously empty scale value. - Compare the new stack height with the two existing stack heights. - After adding the new X, identify which stack or stacks are tallest.

Solution

1. The new X creates a third measured length, so \(3\) different lengths are shown. 2. The original stacks still each have \(3\) Xs, while the new stack has \(1\) X. 3. The greatest number of Xs in any stack remains \(3\).

Answer

The plot shows \(3\) different lengths, and the tallest stacks have \(3\) Xs.
5403923
The line plot below shows nail lengths. Each X represents one nail. One X is moved from \(1\frac{1}{4}\) inches to \(1\frac{1}{2}\) inches. What are the new stack heights, and which length then appears most often?
Figure for problem 540392

Hints

- Remove one X from the old measurement value. - Add one X to the new measurement value. - Compare the three updated stack heights.

Solution

1. The first stack decreases from \(3\) to \(2\) Xs. 2. The second stack increases from \(2\) to \(3\) Xs. 3. The third stack stays at \(1\) X. 4. The length \(1\frac{1}{2}\) inches now appears most often.

Answer

The new stack heights are \(2\), \(3\), and \(1\) Xs. The length \(1\frac{1}{2}\) inches appears most often.
5404023
The line plot below shows ribbon lengths. Each X represents one ribbon. One new X is added at \(3\frac{1}{4}\) inches. Which length then appears the fewest times?
Figure for problem 540402

Hints

- Update the stack that receives the new X. - Count the Xs above each ribbon length. - Find the length with the smallest number of Xs.

Solution

1. The number of Xs at \(3\frac{1}{4}\) inches becomes \(1 + 1 = 2\). 2. The other numbers of Xs are \(3\) and \(1\). 3. The length \(3\frac{3}{4}\) inches appears only once, so it appears the fewest times.

Answer

\(3\frac{3}{4}\) inches appears the fewest times.
5404063
A quarter-inch line-plot scale is labeled from \(2\) inches through \(3\) inches. The data include a measurement of \(3\frac{1}{4}\) inches. What change is needed so every measurement can be plotted?

Hints

- Compare the greatest data value with the current greatest scale label. - Keep the same quarter-inch interval when extending the scale.

Solution

1. The current scale ends before the value \(3\frac{1}{4}\). 2. Extend the scale one quarter-inch interval and add the label \(3\frac{1}{4}\).

Answer

Extend the scale to \(3\frac{1}{4}\) inches and add that label at the next equally spaced mark.
5404173
The line plot below shows craft-stick lengths. Each X represents one craft stick. How many craft sticks are at least \(1\frac{1}{2}\) inches long?
Figure for problem 540417

Hints

- Start at \(1\frac{1}{2}\) inches on the scale. - Include the stack at that value and every stack to its right. - Add the Xs in the included stacks.

Solution

1. “At least \(1\frac{1}{2}\) inches” includes \(1\frac{1}{2}\), \(1\frac{3}{4}\), and \(2\) inches. 2. Add the Xs at those lengths: \(5 + 3 + 1 = 9\).

Answer

There are \(9\) craft sticks that are at least \(1\frac{1}{2}\) inches long.
5404263
A line-plot scale is marked every \(\frac{1}{4}\) inch. An X for \(2\frac{1}{2}\) inches was placed two marks too far to the right. At what label was the X placed, and how many marks must it move to be corrected?

Hints

- Read the quarter-inch labels one mark at a time. - Start at \(2\frac{1}{2}\) inches and count two marks to the right. - Reverse that movement to correct the X.

Solution

1. One mark to the right of \(2\frac{1}{2}\) inches is \(2\frac{3}{4}\) inches. 2. Two marks to the right is \(3\) inches. 3. The X must move two marks to the left to return to \(2\frac{1}{2}\) inches.

Answer

The X was placed at \(3\) inches. It must move \(2\) marks to the left.
5404313
The line plot below shows ribbon lengths. Each X represents one ribbon. a) How many ribbons were measured? b) How many ribbons are longer than \(1\frac{1}{2}\) inches?
Figure for problem 540431

Hints

- Each X represents one measured ribbon. - Add every stack to find the total number measured. - For part b), count only stacks at lengths greater than \(1\frac{1}{2}\) inches.

Solution

1. For part a), add all the Xs: \(2 + 1 + 3 + 2 + 1 = 9\). 2. For part b), use the stacks to the right of \(1\frac{1}{2}\) inches: \(3 + 2 + 1 = 6\).

Answer

a) \(9\) ribbons b) \(6\) ribbons
5404443
The line plot below shows paper-strip lengths. Each X represents one strip. How many measurements are at whole-number lengths, and how many are at lengths that are not whole numbers?
Figure for problem 540444

Hints

- Identify which labels are whole numbers. - Count the Xs at the whole-number label. - Add the Xs at all remaining labels.

Solution

1. Only \(1\) inch is a whole-number length, and it has \(2\) Xs. 2. The other lengths are not whole numbers and have \(1 + 3 + 4 = 8\) Xs altogether.

Answer

There are \(2\) measurements at whole-number lengths and \(8\) at lengths that are not whole numbers.
5404483
The line plot below shows plant-leaf lengths. Each X represents one leaf. How many leaves are longer than \(1\frac{1}{4}\) inches but shorter than \(2\) inches?
Figure for problem 540448

Hints

- Do not include either endpoint named in the question. - Identify the labels strictly between \(1\frac{1}{4}\) and \(2\) inches. - Add the Xs at those labels.

Solution

1. The included lengths are \(1\frac{1}{2}\) inches and \(1\frac{3}{4}\) inches. 2. Add the Xs at those lengths: \(3 + 4 = 7\).

Answer

There are \(7\) leaves in the stated length range.
5404533
The line plot below shows string lengths. Each X represents one string. How many strings are not exactly \(2\) inches long?
Figure for problem 540453

Hints

- Do not count the middle stack at \(2\) inches. - Use the stacks at \(1\frac{1}{2}\) and \(2\frac{1}{2}\) inches. - Add the Xs in those two stacks.

Solution

1. Strings that are not exactly \(2\) inches long are in the two outside stacks. 2. Add those Xs: \(4 + 4 = 8\).

Answer

There are \(8\) strings that are not exactly \(2\) inches long.
5404623
The line plot below shows ribbon lengths. Each X represents one ribbon. How many measurements are below \(1\frac{1}{2}\) inches, exactly \(1\frac{1}{2}\) inches, and above \(1\frac{1}{2}\) inches?
Figure for problem 540462

Hints

- Compare each ribbon length with \(1\frac{1}{2}\) inches. - Keep lengths equal to the reference value in their own group. - Count the values in each of the three groups.

Solution

1. The values below \(1\frac{1}{2}\) are \(1\), \(1\frac{1}{4}\), and \(1\frac{1}{4}\), giving \(3\) measurements. 2. There is \(1\) measurement exactly at \(1\frac{1}{2}\). 3. The values above \(1\frac{1}{2}\) are \(1\frac{3}{4}\), \(2\), and \(2\), giving \(3\) measurements.

Answer

There are \(3\) ribbon lengths below, \(1\) exactly equal to, and \(3\) above \(1\frac{1}{2}\) inches.
5548583
The line plot shows board lengths measured to the nearest quarter inch. How many boards are at least \(2\frac{3}{4}\) inches long?
Figure for problem 554858

Hints

- “At least” includes the stated endpoint. - Count the Xs at \(2\frac{3}{4}\) inches and at every greater labeled value. - Add only those qualifying Xs.

Solution

1. At \(2\frac{3}{4}\) inches, the plot has \(2\) Xs. 2. At \(3\) inches, the plot has \(1\) X. 3. These are the values at least \(2\frac{3}{4}\) inches, so the total is \(2 + 1 = 3\) boards.

Answer

\(3\) boards
5548593
A line plot shows three ribbon lengths. One X was placed at \(3\frac{1}{4}\) inches but should have been placed at \(3\frac{1}{2}\) inches. After correcting that one X, how many Xs will be in each stack, and will there be one length that occurs most often?
Figure for problem 554859

Hints

- Read the three original stack heights from the line plot. - Moving one X decreases one stack by \(1\) and increases another by \(1\). - Compare the corrected stack heights to decide whether one is tallest.

Solution

1. Before the correction, the stack heights are \(3\) at \(3\frac{1}{4}\), \(1\) at \(3\frac{1}{2}\), and \(2\) at \(3\frac{3}{4}\). 2. Move one X from \(3\frac{1}{4}\) to \(3\frac{1}{2}\). 3. The corrected stack heights are \(2, 2, 2\). 4. Since all three stacks are equal, there is no single length that occurs most often.

Answer

Each stack will have \(2\) Xs. There will be no single most frequent length.
5403013
Two classes made the line plots below for pencil lengths. Each X represents one pencil. If the data are combined, how many Xs are at each length, and which length appears more often?
Figure for problem 540301

Hints

- Combine the Xs at matching measurement values from the two plots. - Keep the two length values separate while adding their stack heights. - Compare the new stack heights to see which length appears more often.

Solution

1. At \(2\frac{1}{4}\) inches, combine \(2 + 1 = 3\) Xs. 2. At \(2\frac{1}{2}\) inches, combine \(1 + 3 = 4\) Xs. 3. Since \(4 > 3\), \(2\frac{1}{2}\) inches appears more often.

Answer

The combined plot has \(3\) Xs at \(2\frac{1}{4}\) inches and \(4\) Xs at \(2\frac{1}{2}\) inches. The length \(2\frac{1}{2}\) inches appears more often.
5404123
The line plot below shows paper-strip lengths. Each X represents one strip. All strips measuring \(2\frac{1}{4}\) inches are removed. How many strips remain, and how many different lengths remain?
Figure for problem 540412

Hints

- Count all the Xs before removing any strips. - Remove the entire stack at \(2\frac{1}{4}\) inches. - Then count both the remaining Xs and the lengths that still have at least one X.

Solution

1. The original plot has \(2 + 3 + 1 + 2 = 8\) Xs. 2. Removing the \(3\) Xs at \(2\frac{1}{4}\) inches leaves \(8 - 3 = 5\) Xs. 3. Xs remain at \(2\), \(2\frac{1}{2}\), and \(2\frac{3}{4}\) inches, so \(3\) different lengths remain.

Answer

There are \(5\) strips left at \(3\) different lengths.
5404353
A quarter-inch line-plot scale ends at \(3\) inches and has \(9\) marks including both endpoints. Theo says there are \(9\) quarter-inch intervals, so the first label is \(\frac{3}{4}\) inch. Explain the off-by-one error and find the correct first label.

Hints

- Compare the number of marks with the number of spaces between adjacent marks. - Group the quarter-inch intervals into sets of four to make whole inches. - Work backward from the \(3\)-inch endpoint by the total span of the intervals.

Solution

1. Nine marks have only \(8\) spaces between them, so there are \(8\) intervals. 2. Four quarter-inch intervals make \(1\) inch, so eight quarter-inch intervals make \(2\) inches. 3. Work back \(2\) inches from \(3\) inches: \(3 - 2 = 1\) inch.

Answer

Theo counted marks instead of intervals. The first label is \(1\) inch.
5404493
The line plot below shows wooden-piece lengths. Each X represents one piece. How many more pieces are at least \(1\frac{1}{2}\) inches long than are at most \(1\frac{1}{4}\) inches long?
Figure for problem 540449

Hints

- Form the “at least” group using the two greatest labels. - Form the “at most” group using the two least labels. - Compare the two group totals by subtraction.

Solution

1. At least \(1\frac{1}{2}\) inches includes \(8 + 3 = 11\) pieces. 2. At most \(1\frac{1}{4}\) inches includes \(3 + 4 = 7\) pieces. 3. Find the difference: \(11 - 7 = 4\).

Answer

There are \(4\) more pieces in the group that is at least \(1\frac{1}{2}\) inches long.
5404513
A line plot of ribbon lengths has quarter-inch marks from \(2\) inches through \(4\) inches. Measurements occur only at every other mark, beginning at \(2\) inches and ending at \(4\) inches. List all labels that can have Xs and state how many such labels there are.

Hints

- Skipping one quarter-inch mark means moving two quarter-inch intervals each time. - Continue the pattern from the first endpoint to the last endpoint. - Count the listed labels, including both endpoints.

Solution

1. Every other quarter-inch mark is separated by \(\frac{1}{2}\) inch. 2. Starting at \(2\), the labels are \(2\), \(2\frac{1}{2}\), \(3\), \(3\frac{1}{2}\), and \(4\). 3. There are \(5\) such labels.

Answer

The labels are \(2\), \(2\frac{1}{2}\), \(3\), \(3\frac{1}{2}\), and \(4\) inches. There are \(5\) labels.
5404653
A line plot shows the lengths of plant stems. Its quarter-inch scale runs from \(1\) inch through \(3\) inches. Every scale mark has exactly one X except \(1\frac{1}{4}\) inches and \(2\frac{3}{4}\) inches, which have no Xs. How many stems are shown, and what lengths do they have?

Hints

- List every quarter-inch mark between the endpoints, including both endpoints. - Remove only the two labels stated to have no Xs. - Each remaining X represents one measured stem.

Solution

1. The quarter-inch labels are \(1\), \(1\frac{1}{4}\), \(1\frac{1}{2}\), \(1\frac{3}{4}\), \(2\), \(2\frac{1}{4}\), \(2\frac{1}{2}\), \(2\frac{3}{4}\), and \(3\). 2. There are \(9\) scale marks, and \(2\) have no X, so the plot has \(9 - 2 = 7\) Xs. 3. The represented lengths are \(1\), \(1\frac{1}{2}\), \(1\frac{3}{4}\), \(2\), \(2\frac{1}{4}\), \(2\frac{1}{2}\), and \(3\) inches.

Answer

The plot shows \(7\) stems. Their lengths are \(1\), \(1\frac{1}{2}\), \(1\frac{3}{4}\), \(2\), \(2\frac{1}{4}\), \(2\frac{1}{2}\), and \(3\) inches.
5404683
The line plot below shows ribbon lengths in yards. Each X represents one ribbon. a) How many ribbons are shown? b) Which length appears most often? c) How many ribbons are at least \(1\frac{1}{2}\) yards long?
Figure for problem 540468

Hints

- Add all the X counts to find the total. - The length with the tallest stack appears most often. - “At least \(1\frac{1}{2}\) yards” includes that length and all greater lengths.

Solution

1. For part a), add the X counts: \(1 + 2 + 2 + 4 + 1 + 2 = 12\). 2. For part b), the tallest stack has \(4\) Xs at \(1\frac{1}{2}\) yards. 3. For part c), add the counts at \(1\frac{1}{2}\), \(1\frac{3}{4}\), and \(2\) yards: \(4 + 1 + 2 = 7\).

Answer

a) \(12\) ribbons b) \(1\frac{1}{2}\) yards c) \(7\) ribbons
5548603
Two line plots show nail lengths measured to the nearest quarter inch. Which plot has more nails that are at least \(4\frac{3}{4}\) inches long, and by how many nails?
Figure for problem 554860

Hints

- “At least” includes \(4\frac{3}{4}\) inches and all greater values. - Count qualifying Xs in each plot separately. - Compare the two totals with subtraction.

Solution

1. In plot a), there are \(2\) Xs at \(4\frac{3}{4}\) inches and none at \(5\) inches, for \(2\) qualifying nails. 2. In plot b), there is \(1\) X at \(4\frac{3}{4}\) inches and \(2\) Xs at \(5\) inches, for \(3\) qualifying nails. 3. Plot b) has \(3 - 2 = 1\) more qualifying nail.

Answer

Plot b) has more nails that are at least \(4\frac{3}{4}\) inches long, by \(1\) nail.
5548613
Mina measured six pencil lengths to the nearest quarter inch and recorded \(5, 5\frac{1}{2}, 6\frac{1}{4}, 5\frac{3}{4}, 5\frac{1}{2}, 6\frac{1}{4}\) inches. A line plot must run from \(5\) inches through \(6\frac{1}{4}\) inches in quarter-inch intervals. List every scale label, including labels with no measurements, and state the number of Xs above each label. Explain why the empty labels must remain on the scale.

Hints

- Build the scale first by counting in quarter-inch steps from the lower endpoint to the upper endpoint. - Tally each measured pencil at its matching scale value. - A scale value can have zero Xs and still be necessary for equal spacing.

Solution

1. Quarter-inch steps from \(5\) to \(6\frac{1}{4}\) are \(5, 5\frac{1}{4}, 5\frac{1}{2}, 5\frac{3}{4}, 6, 6\frac{1}{4}\). 2. The counts are \(1, 0, 2, 1, 0, 2\) in that order. 3. The labels at \(5\frac{1}{4}\) and \(6\) stay on the scale even though they have no Xs, because removing them would break the equal quarter-inch spacing.

Answer

Scale and X counts: \(5\): \(1\) X \(5\frac{1}{4}\): \(0\) Xs \(5\frac{1}{2}\): \(2\) Xs \(5\frac{3}{4}\): \(1\) X \(6\): \(0\) Xs \(6\frac{1}{4}\): \(2\) Xs The empty labels must remain so the scale keeps equal quarter-inch intervals.

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