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Compare fractions same denominator

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5401403
A bar is divided into \(6\) equal parts. Some parts are shaded. What fraction is shaded? What fraction is unshaded? Compare the two fractions.
Figure for problem 540140

Hints

- Count the shaded parts and all the equal parts. - Count the unshaded parts. - Compare the numerators because the denominators are the same.

Solution

1. Three of the \(6\) equal parts are shaded, so the shaded fraction is \(\frac{3}{6}\). 2. The other \(3\) parts are unshaded, so the unshaded fraction is \(\frac{3}{6}\). 3. The fractions are equal: \(\frac{3}{6}=\frac{3}{6}\).

Answer

Shaded: \(\frac{3}{6}\) Unshaded: \(\frac{3}{6}\) \(\frac{3}{6}=\frac{3}{6}\)
5402033
Order these fractions from least to greatest: \(\frac{5}{8},\ \frac{1}{8},\ \frac{7}{8}\)

Hints

- The denominator is \(8\) in every fraction. - Compare the numerators \(1\), \(5\), and \(7\). - Put the smallest numerator first.

Solution

1. All three fractions have denominator \(8\), so compare their numerators. 2. Since \(1<5<7\), the fractions are ordered \(\frac{1}{8}<\frac{5}{8}<\frac{7}{8}\).

Answer

\(\frac{1}{8}<\frac{5}{8}<\frac{7}{8}\)
5401103
Consider the fractions \(\frac{4}{6}\), \(\frac{6}{6}\), and \(\frac{7}{6}\). Label each fraction “less than \(1\),” “equal to \(1\),” or “greater than \(1\).” Then write the fractions from least to greatest.

Hints

- First name one whole using sixths. - Compare each numerator with the denominator. - Use the comparisons with \(1\) to determine the order.

Solution

1. One whole is \(\frac{6}{6}\). 2. Since \(4<6\), \(\frac{4}{6}\) is less than \(1\). 3. \(\frac{6}{6}\) is equal to \(1\), and since \(7>6\), \(\frac{7}{6}\) is greater than \(1\). 4. Their order is \(\frac{4}{6}<\frac{6}{6}<\frac{7}{6}\).

Answer

Less than \(1\): \(\frac{4}{6}\) Equal to \(1\): \(\frac{6}{6}\) Greater than \(1\): \(\frac{7}{6}\) \(\frac{4}{6}<\frac{6}{6}<\frac{7}{6}\)
5401113
List every fraction with denominator \(8\) that is greater than \(\frac{3}{8}\) and less than \(\frac{6}{8}\).

Hints

- Keep the denominator \(8\) in every fraction. - Find whole numbers greater than \(3\) and less than \(6\). - Use those numbers as the numerators.

Solution

1. With denominator \(8\), compare the numerators because every part has the same size. 2. The numerator must be greater than \(3\) and less than \(6\). 3. The whole-number numerators that fit are \(4\) and \(5\). 4. The fractions are \(\frac{4}{8}\) and \(\frac{5}{8}\).

Answer

\(\frac{4}{8}\) and \(\frac{5}{8}\)
5401163
Two same-size badges are each divided into \(6\) equal parts. Badge a) has \(2\) parts shaded. Badge b) has \(5\) parts shaded. Which badge has the greater unshaded fraction? Write and compare the unshaded fractions.
Figure for problem 540116

Hints

- Find the unshaded count in each six-part whole. - Keep the denominator tied to the total equal parts. - Compare the resulting numerators.

Solution

1. Badge a) has \(6-2=4\) unshaded parts, so its unshaded fraction is \(\frac{4}{6}\). 2. Badge b) has \(6-5=1\) unshaded part, so its unshaded fraction is \(\frac{1}{6}\). 3. The parts are the same size, and \(4>1\), so \(\frac{4}{6}>\frac{1}{6}\).

Answer

Badge a) has the greater unshaded fraction: \(\frac{4}{6}>\frac{1}{6}\).
5401193
Two same-size snack trays each have \(8\) equal spaces. One tray has \(5\) spaces filled, and the other has \(2\) spaces filled. Which filled fraction is greater? Write a comparison.

Hints

- Both trays are divided into \(8\) equal spaces. - Write the filled part of each tray as a fraction. - With equal denominators, the greater numerator names the greater fraction.

Solution

1. The filled fractions are \(\frac{5}{8}\) and \(\frac{2}{8}\). 2. The denominators are the same, so compare the numerators. 3. Since \(5>2\), \(\frac{5}{8}>\frac{2}{8}\).

Answer

\(\frac{5}{8}>\frac{2}{8}\)
5401343
Two same-size bars are each divided into \(8\) equal parts. Each bar has \(4\) parts shaded, but the shaded parts are in different positions. Compare the shaded fractions with \(<\), \(>\), or \(=\). Explain.
Figure for problem 540134

Hints

- Count the equal parts and shaded parts in each bar. - The shaded pieces can be in different places and still have the same total size. - Use \(=\) when the shaded amounts are equal.

Solution

1. Each bar has the same denominator, \(8\), so its parts are the same size. 2. Each bar also has the same numerator, \(4\). 3. The placement of the shaded parts does not change their total amount. 4. Therefore, \(\frac{4}{8}=\frac{4}{8}\).

Answer

\(\frac{4}{8}=\frac{4}{8}\)
5401383
Ava walks \(\frac{6}{4}\) mile on one trail. Ben walks \(\frac{3}{4}\) mile on another trail. Who walks farther? Write a fraction comparison.

Hints

- Both fractions have denominator \(4\). - Compare the number of fourths in each distance. - The greater numerator names the greater distance.

Solution

1. Both distances are measured in fourths of a mile. 2. Compare the numerators \(6\) and \(3\). 3. Since \(6>3\), \(\frac{6}{4}>\frac{3}{4}\). 4. Ava walks farther.

Answer

Ava walks farther. \(\frac{6}{4}>\frac{3}{4}\).
5401693
A student orders these fractions from least to greatest: \(\frac{10}{8},\ \frac{7}{8},\ \frac{9}{8}\) Find the error and write the correct order.

Hints

- Identify what stays the same in all three fractions. - Think of each numerator as a count of same-size pieces. - Compare each fraction with one whole, \(\frac{8}{8}\).

Solution

1. All three fractions count eighth-size parts. 2. Their numerators are \(10\), \(7\), and \(9\). 3. Ordering the numerators gives \(7<9<10\). 4. Therefore, \(\frac{7}{8}<\frac{9}{8}<\frac{10}{8}\).

Answer

The student did not order the numerators. The correct order is \(\frac{7}{8}<\frac{9}{8}<\frac{10}{8}\).
5401833
A recipe uses \(\frac{5}{8}\) cup of oats and \(\frac{7}{8}\) cup of fruit. Which amount is greater, and by how many eighths?

Hints

- Both fractions have denominator \(8\). - Compare the numerators. - Subtract the smaller numerator from the larger numerator to find the difference in eighths.

Solution

1. Both amounts are measured in eighths of a cup. 2. Since \(7>5\), \(\frac{7}{8}>\frac{5}{8}\). 3. The difference is \(7-5=2\) eighths.

Answer

The fruit amount is greater: \(\frac{7}{8}>\frac{5}{8}\). It is greater by \(2\) eighths.
5402083
One container is \(\frac{2}{8}\) full, and another same-size container is \(\frac{6}{8}\) full. Which container is fuller, and by how many eighths?

Hints

- Both containers are divided into eighths. - Compare the numerators. - Subtract the smaller numerator from the larger numerator.

Solution

1. Both fractions have denominator \(8\), so compare the numerators. 2. Since \(6>2\), \(\frac{6}{8}>\frac{2}{8}\). 3. The difference is \(6-2=4\) eighths.

Answer

The second container is fuller: \(\frac{6}{8}>\frac{2}{8}\). It is fuller by \(4\) eighths.
5402113
Order these fractions from least to greatest: \(\frac{7}{4},\ \frac{5}{4},\ \frac{6}{4}\)

Hints

- The denominator is \(4\) in every fraction. - Compare the numerators \(5\), \(6\), and \(7\). - Put the smallest numerator first.

Solution

1. All three fractions have denominator \(4\), so compare their numerators. 2. Since \(5<6<7\), the order is \(\frac{5}{4}<\frac{6}{4}<\frac{7}{4}\).

Answer

\(\frac{5}{4}<\frac{6}{4}<\frac{7}{4}\)
5402233
Point \(F\) is one sixth-size space to the left of \(1\). Write the fraction at point \(F\) and compare it with \(\frac{2}{6}\).

Hints

- Write \(1\) as sixths. - Move left by one sixth-size space. - Compare the numerators because both fractions have denominator \(6\).

Solution

1. One whole is \(\frac{6}{6}\). 2. One sixth-size space left of \(\frac{6}{6}\) is \(\frac{5}{6}\). 3. Since \(5>2\), \(\frac{5}{6}>\frac{2}{6}\).

Answer

\(F=\frac{5}{6}\), and \(\frac{5}{6}>\frac{2}{6}\).
5402273
Fraction A is \(\frac{7}{8}\). Fraction B has the same denominator and is one eighth-size part greater. Find fraction B and compare it with A and with \(1\).

Hints

- Add one eighth while keeping the denominator \(8\). - Compare the new numerator with the old numerator. - A fraction equals one whole when its numerator equals its denominator.

Solution

1. One eighth greater than \(\frac{7}{8}\) is \(\frac{8}{8}\). 2. Fraction B is \(\frac{8}{8}\). 3. Since \(8>7\), \(\frac{8}{8}>\frac{7}{8}\). 4. Also, \(\frac{8}{8}=1\).

Answer

Fraction B is \(\frac{8}{8}\). Thus, \(\frac{7}{8}<\frac{8}{8}=1\).
5401463
Two same-size trays are each divided into \(6\) equal spaces. Tray A has \(3\) spaces filled. Tray B has the same number of empty spaces as Tray A has filled spaces. Compare the filled fractions of the two trays.

Hints

- Find both the filled and empty counts for Tray A. - Use the stated relationship to determine Tray B's empty count. - Convert Tray B's empty count into a filled count before comparing.

Solution

1. Tray A has \(3\) filled spaces and \(6-3=3\) empty spaces. 2. Tray B has \(3\) empty spaces, so it also has \(6-3=3\) filled spaces. 3. Both filled fractions are \(\frac{3}{6}\). 4. Therefore, the filled fractions are equal.

Answer

\(\frac{3}{6}=\frac{3}{6}\)
5401483
Compare \(\frac{7}{8}\) and \(\frac{9}{8}\). Which fraction is closer to \(1\), or are they equally close? Also write the correct comparison symbol.

Hints

- One whole is \(\frac{8}{8}\). - Count how many eighth-size spaces each fraction is from \(\frac{8}{8}\). - Then compare the numerators to order the fractions.

Solution

1. One whole is \(\frac{8}{8}\). 2. Seven eighths is one eighth below one, and nine eighths is one eighth above one. 3. The fractions are equally close to one. 4. Since \(7<9\), \(\frac{7}{8}<\frac{9}{8}\).

Answer

They are equally close to \(1\), and \(\frac{7}{8}<\frac{9}{8}\).
5401593
Compare \(\frac{5}{6}\) and \(\frac{8}{6}\). Which fraction is closer to \(1\)?

Hints

- One whole is \(\frac{6}{6}\). - Count how many sixth-size spaces each fraction is from \(\frac{6}{6}\). - Then compare the numerators to order the fractions.

Solution

1. One whole is \(\frac{6}{6}\). 2. Five sixths is \(1\) sixth below one, while eight sixths is \(2\) sixths above one. 3. Therefore, \(\frac{5}{6}\) is closer to \(1\). 4. Since \(5<8\), \(\frac{5}{6}<\frac{8}{6}\).

Answer

\(\frac{5}{6}<\frac{8}{6}\), and \(\frac{5}{6}\) is closer to \(1\).
5401723
Two eighths fractions change. Fraction A changes from \(\frac{2}{8}\) to \(\frac{5}{8}\). Fraction B changes from \(\frac{4}{8}\) to \(\frac{6}{8}\). Which final fraction is greater? Which fraction increased by more eighth-size parts?

Hints

- Treat the final comparison and the amount of change as two separate questions. - The shared denominator means every counted part has the same size. - Compare ending numerators, then compare how much each numerator changed.

Solution

1. The final numerators are \(5\) and \(6\), so \(\frac{6}{8}>\frac{5}{8}\). 2. Fraction A gains \(5-2=3\) eighth-size parts. 3. Fraction B gains \(6-4=2\) eighth-size parts. 4. Fraction B ends greater, but fraction A increases by more parts.

Answer

Fraction B ends greater: \(\frac{6}{8}>\frac{5}{8}\). Fraction A increases by more, gaining \(3\) eighths instead of \(2\) eighths.
5401773
A light bar has \(8\) equal sections. It starts with \(5\) sections lit. Then \(2\) more sections light up, and after that \(1\) lit section turns off. Write the fraction lit at each stage. Order the three fractions from least to greatest.

Hints

- Keep the denominator fixed because the bar still has eight sections. - Update only the number of lit sections at each change. - Order the resulting fractions by their numerators.

Solution

1. The starting fraction is \(\frac{5}{8}\). 2. After \(2\) more sections light up, the fraction is \(\frac{7}{8}\). 3. After \(1\) section turns off, the fraction is \(\frac{6}{8}\). 4. Since \(5<6<7\), \(\frac{5}{8}<\frac{6}{8}<\frac{7}{8}\).

Answer

The stages are \(\frac{5}{8}\), \(\frac{7}{8}\), and \(\frac{6}{8}\). From least to greatest: \(\frac{5}{8}<\frac{6}{8}<\frac{7}{8}\).
5401923
Eight voting cards are arranged in \(2\) identical groups. Each group has \(3\) “yes” cards and \(1\) “no” card. What fraction of all cards say “yes”? Compare that fraction with \(\frac{1}{2}\) and with \(1\).

Hints

- Count the “yes” cards in both groups. - Put that count over the \(8\) cards in all. - Write one half and one whole as eighths before comparing.

Solution

1. Two groups contain \(3+3=6\) yes cards. 2. There are \(8\) cards in all, so the yes fraction is \(\frac{6}{8}\). 3. One half is \(\frac{4}{8}\), and one whole is \(\frac{8}{8}\). 4. Therefore, \(\frac{1}{2}<\frac{6}{8}<1\).

Answer

The yes fraction is \(\frac{6}{8}\), and \(\frac{1}{2}<\frac{6}{8}<1\).
5401953
A progress bar shows \(\frac{4}{8}\) complete. A second bar shows the first bar's completed eighths plus half of the first bar's remaining eighths. What fraction does the second bar show? Compare the two bars.

Hints

- Find the uncompleted part of the first bar. - Take half of that remaining count. - Add it to the original completed count before comparing.

Solution

1. The first bar has \(8-4=4\) eighths remaining. 2. Half of the \(4\) remaining eighths is \(2\) eighths. 3. The second bar shows \(4+2=6\) eighths, or \(\frac{6}{8}\). 4. Since \(6>4\), \(\frac{6}{8}>\frac{4}{8}\).

Answer

The second bar shows \(\frac{6}{8}\), and \(\frac{6}{8}>\frac{4}{8}\).
5401993
Fraction A is \(\frac{2}{6}\). Fraction B also has denominator \(6\) and has the same number of unshaded sixths as fraction A has shaded sixths. Find fraction B and compare the fractions.

Hints

- Fraction A has \(2\) shaded sixths. - Give Fraction B \(2\) unshaded sixths, then count its shaded sixths. - Compare the numerators because both denominators are \(6\).

Solution

1. Fraction A has \(2\) shaded sixths. 2. Fraction B therefore has \(2\) unshaded sixths. 3. Out of \(6\) parts, that leaves \(6-2=4\) shaded sixths, so B is \(\frac{4}{6}\). 4. Since \(4>2\), \(\frac{4}{6}>\frac{2}{6}\).

Answer

Fraction B is \(\frac{4}{6}\), and \(\frac{4}{6}>\frac{2}{6}\).
5402063
Consider \(\frac{4}{6},\ \frac{5}{6},\) and \(\frac{7}{6}\). Order them from least to greatest. Which pair is closest together, and how many sixth-size parts separate each neighboring pair in the order?

Hints

- All three fractions are measured in sixths. - Order the numerators first. - Compare the differences between neighboring numerators to find the closest pair.

Solution

1. Ordering the numerators gives \(4<5<7\), so \(\frac{4}{6}<\frac{5}{6}<\frac{7}{6}\). 2. The first neighboring pair differs by \(5-4=1\) sixth-size part. 3. The second neighboring pair differs by \(7-5=2\) sixth-size parts. 4. Therefore, \(\frac{4}{6}\) and \(\frac{5}{6}\) are the closest pair.

Answer

\(\frac{4}{6}<\frac{5}{6}<\frac{7}{6}\). The first pair is \(1\) sixth apart, the second pair is \(2\) sixths apart, and \(\frac{4}{6}\) with \(\frac{5}{6}\) is closest.
5402163
Fraction A is \(\frac{3}{4}\), and fraction B is \(\frac{5}{4}\). Two fourth-size parts are added to each fraction. Compare A and B before the change and compare the new fractions after the change. Does the order change?

Hints

- Compare the original equal-part counts first. - Apply the same numerator change to both fractions. - Check whether the new numerator order matches the original order.

Solution

1. Initially, \(3<5\), so \(\frac{3}{4}<\frac{5}{4}\). 2. Adding \(2\) fourths changes A to \(\frac{5}{4}\). 3. Adding \(2\) fourths changes B to \(\frac{7}{4}\). 4. Since \(5<7\), \(\frac{5}{4}<\frac{7}{4}\), so the order does not change.

Answer

Before: \(\frac{3}{4}<\frac{5}{4}\). After: \(\frac{5}{4}<\frac{7}{4}\). The order stays the same.
5401663
Panel A has \(8\) lights, with \(\frac{7}{8}\) of them on. Panel B also has \(8\) lights, with an unknown number on. If \(2\) lights are switched off in Panel A and \(2\) lights are switched on in Panel B, both panels show \(\frac{5}{8}\) of their lights on. What fraction of Panel B's lights were on at first? Compare the starting fractions.

Hints

- Work backward from the equal ending fractions. - Keep the total number of lights in each panel fixed while changing how many are on. - Compare the original numerators only after finding the missing one.

Solution

1. Panel B gains \(2\) lights that are on and ends at \(\frac{5}{8}\). 2. Its starting numerator was \(5-2=3\), so it started at \(\frac{3}{8}\). 3. Both starting fractions use eighths, and \(7>3\). 4. Therefore, \(\frac{7}{8}>\frac{3}{8}\).

Answer

Panel B started at \(\frac{3}{8}\). The starting comparison is \(\frac{7}{8}>\frac{3}{8}\).

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