Fractions and Mixed Numbers: Questions and Step-by-Step Solutions - Grade 6

The large square consists of 16 equal small squares.

Red: There are 4 red squares. The fraction is: $$\dfrac{4}{16} = \dfrac{1}{4}$$

Blue: There is 1 blue square. The fraction is: $$\dfrac{1}{16}$$

Orange: There is half of a square that is orange. The fraction is: $$\dfrac{1}{2} \times \dfrac{1}{16} = \dfrac{1}{32}$$

Green: There are one and a half green squares. The fraction is: $$\dfrac{1}{16} + \left(\dfrac{1}{2} \times \dfrac{1}{16}\right) = \dfrac{1}{16} + \dfrac{1}{32} = \dfrac{2}{32} + \dfrac{1}{32} = \dfrac{3}{32}$$

Black: There are 3 black squares. The fraction is: $$\dfrac{3}{16}$$

Yellow: There are 3 yellow squares. The fraction is: $$\dfrac{3}{16}$$

Correct Answer: D

There are two whole items and \(\dfrac{3}{4}\) of one item shaded.

Hence, the shaded part is: $$2 + \dfrac{3}{4} = 2 \dfrac{3}{4}$$

Correct Answer: C

Note that \(1 \dfrac{1}{5}\) is larger than 1 and smaller than 2.

Each small division on the number line represents: $$\dfrac{1}{10}$$

Hence, point R represents: $$1 + \dfrac{2}{10} = 1 \dfrac{2}{10}$$

Simplifying the fraction: $$1 \dfrac{2}{10} = 1 \dfrac{1}{5}$$

Therefore, point R represents \(1 \dfrac{1}{5}\).

Correct Answer: B

To add the mixed numbers, combine the whole numbers and the fractional parts separately:

$$3 \dfrac{1}{2} + 5 \dfrac{1}{2} = (3 + 5) + \left(\dfrac{1}{2} + \dfrac{1}{2}\right)$$

$$= 8 + \dfrac{2}{2} = 8 + 1 = 9$$

Correct Answer: C

To add fractions, first write them with a common denominator:

$$\dfrac{1}{2} + \dfrac{1}{14} = \dfrac{7}{14} + \dfrac{1}{14}$$

Now add the numerators:

$$\dfrac{7}{14} + \dfrac{1}{14} = \dfrac{8}{14}$$

Finally, simplify the fraction by dividing the numerator and denominator by 2:

$$\dfrac{8}{14} = \dfrac{4}{7}$$

Correct Answer: A

To subtract fractions, first write them with a common denominator:

$$\dfrac{1}{3} - \dfrac{1}{12} = \dfrac{4}{12} - \dfrac{1}{12}$$

Now subtract the numerators:

$$\dfrac{4}{12} - \dfrac{1}{12} = \dfrac{3}{12}$$

Finally, simplify the fraction by dividing the numerator and denominator by 3:

$$\dfrac{3}{12} = \dfrac{1}{4}$$

Correct Answer: B

One half is written as: $$\dfrac{1}{2}$$

Two quarters is written as: $$\dfrac{2}{4}$$

Divide numerator and denominator to reduce fraction: $$\dfrac{2}{4} = \dfrac{1}{2}$$

Hence one half is the same as two quarters.

Correct Answer: B

$$\dfrac{2}{4} = \dfrac{1}{2}$$ (after dividing numerator and denominator by 2)

$$\dfrac{4}{3} = \dfrac{8}{6}$$ (after multiplying numerator and denominator by 2)

$$\dfrac{1}{5} = \dfrac{3}{15}$$ (after multiplying numerator and denominator by 3)

However, \(\dfrac{2}{3}\) and \(\dfrac{8}{9}\) cannot be reduced to each other. They are not equivalent.

Correct Answer: D

To add the mixed numbers, combine the whole numbers and then add the fractional parts:

$$5 \dfrac{2}{3} + 5 \dfrac{1}{2} = (5 + 5) + \left(\dfrac{2}{3} + \dfrac{1}{2}\right)$$

Find a common denominator for the fractions:

$$\dfrac{2}{3} + \dfrac{1}{2} = \dfrac{4}{6} + \dfrac{3}{6} = \dfrac{7}{6}$$

Now substitute back:

$$10 + \dfrac{7}{6} = 10 + \left(\dfrac{6}{6} + \dfrac{1}{6}\right) = 10 + 1 + \dfrac{1}{6} = 11 \dfrac{1}{6}$$

Correct Answer: C

It is easier to compare fractions when they have the same denominator. We choose 18 as the common denominator and rewrite each fraction:

$$\dfrac{8}{9} = \dfrac{16}{18}, \quad \dfrac{17}{18} = \dfrac{17}{18}, \quad \dfrac{2}{3} = \dfrac{12}{18}, \quad \dfrac{7}{6} = \dfrac{21}{18}$$

Now, order the numerators from least to greatest:

$$\dfrac{12}{18} \lt \dfrac{16}{18} \lt \dfrac{17}{18} \lt \dfrac{21}{18}$$

Therefore, the correct original order is: $$\dfrac{2}{3}, \quad \dfrac{8}{9}, \quad \dfrac{17}{18}, \quad \dfrac{7}{6}$$

Correct Answer: B

To compare the fractions, we write them with a common denominator of 110. We then see which one is closest to \(\dfrac{110}{110} = 1\).

$$\dfrac{10}{11} = \dfrac{100}{110}, \quad \dfrac{11}{10} = \dfrac{121}{110}, \quad \dfrac{9}{11} = \dfrac{90}{110}, \quad -\dfrac{9}{10} = -\dfrac{99}{110}$$

Now, since they share the same denominator, we look for the numerator closest to 110:

• \(100\) is 10 away from 110.
• \(121\) is 11 away from 110.
• \(90\) is 20 away from 110.
• \(-99\) is far below 110.

The fraction with the numerator closest to 110 is: $$\dfrac{10}{11}$$

Correct Answer: A

To divide fractions, multiply the first fraction by the reciprocal of the second:

$$\dfrac{5}{2} \div \dfrac{2}{5} = \dfrac{5}{2} \times \dfrac{5}{2}$$

Now multiply the numerators and denominators:

$$\dfrac{5}{2} \times \dfrac{5}{2} = \dfrac{25}{4}$$

Convert the improper fraction to a mixed number:

$$\dfrac{25}{4} = \dfrac{24}{4} + \dfrac{1}{4} = 6 \dfrac{1}{4}$$

Correct Answer: D

To divide a whole number by a fraction, multiply the whole number by the reciprocal of the fraction:

$$5 \div \dfrac{1}{5} = 5 \times \dfrac{5}{1}$$

Now perform the multiplication:

$$5 \times \dfrac{5}{1} = \dfrac{25}{1} = 25$$

Correct Answer: B

To multiply fractions, multiply the numerators together and the denominators together:

$$\dfrac{2}{5} \times \dfrac{7}{8} = \dfrac{2 \times 7}{5 \times 8}$$

Simplify before multiplying where possible (divide 2 and 8 by 2):

$$\dfrac{1 \times 7}{5 \times 4} = \dfrac{7}{20}$$

Correct Answer: C

To convert a mixed number into an improper fraction, multiply the whole number by the denominator, then add the numerator:

$$7 \dfrac{7}{8} = \dfrac{7 \times 8}{8} + \dfrac{7}{8}$$

$$= \dfrac{56}{8} + \dfrac{7}{8} = \dfrac{56 + 7}{8} = \dfrac{63}{8}$$

Correct Answer: D

To convert an improper fraction to a mixed number, divide the numerator by the denominator:

$$\dfrac{31}{8} = \dfrac{24 + 7}{8} = \dfrac{24}{8} + \dfrac{7}{8}$$

$$\dfrac{24}{8} = 3, \quad \text{so we get: } 3 \dfrac{7}{8}$$

Correct Answer: A

To multiply a whole number by a fraction, first write the whole number as a fraction with denominator 1:

$$3 \times \dfrac{1}{4} = \dfrac{3}{1} \times \dfrac{1}{4}$$

Multiply the numerators and denominators:

$$\dfrac{3 \times 1}{1 \times 4} = \dfrac{3}{4}$$

Correct Answer: B

First, convert the mixed numbers into improper fractions:

$$3 \dfrac{1}{4} = \dfrac{13}{4}, \qquad 5 \dfrac{1}{3} = \dfrac{16}{3}$$

Now, divide the fractions by multiplying the first fraction by the reciprocal of the second:

$$\dfrac{13}{4} \div \dfrac{16}{3} = \dfrac{13}{4} \times \dfrac{3}{16}$$

Multiply the numerators and denominators:

$$\dfrac{13 \times 3}{4 \times 16} = \dfrac{39}{64}$$

Correct Answer: D

First, convert the mixed numbers into improper fractions:

$$4 \dfrac{2}{7} = \dfrac{30}{7}, \qquad 5 \dfrac{3}{5} = \dfrac{28}{5}$$

Now multiply the fractions:

$$\dfrac{30}{7} \times \dfrac{28}{5} = \dfrac{30 \times 28}{7 \times 5}$$

Simplify before multiplying (divide 30 and 5 by 5; divide 28 and 7 by 7):

$$\dfrac{6 \times 4}{1 \times 1} = 24$$

Correct Answer: A

We want to solve for \(F\) in the equation: $$F + 2 \dfrac{5}{7} = 4$$

Subtract \(2 \dfrac{5}{7}\) from both sides:

$$F = 4 - 2 \dfrac{5}{7}$$

Borrow 1 from the 4 to create a fraction:

$$F = 3 \dfrac{7}{7} - 2 \dfrac{5}{7} = (3 - 2) + \left(\dfrac{7}{7} - \dfrac{5}{7}\right) = 1 \dfrac{2}{7}$$

Correct Answer: C

We first convert the running times for each day into minutes:

• Monday: \(\dfrac{3}{4} \times 60 = \dfrac{180}{4} = 45\) minutes
• Tuesday: \(30\) minutes
• Wednesday: Half an hour = \(\dfrac{1}{2} \times 60 = 30\) minutes
• Thursday: \(1 \dfrac{1}{4}\) hours = \(60 + \left(\dfrac{1}{4} \times 60\right) = 60 + 15 = 75\) minutes
• Friday: \(\dfrac{2}{3} \times 60 = 40\) minutes

Now add up the total minutes:

$$45 + 30 + 30 + 75 + 40 = 220 \text{ minutes}$$

Convert back to hours and minutes (since 180 minutes = 3 hours):

$$220 = 180 + 40 = 3 \text{ hours and } 40 \text{ minutes}$$

Correct Answer: D

When ordering mixed numbers, we first compare the whole number parts. The number with the smallest whole number will be the smallest overall.

• \(3 \dfrac{1}{5}\) has a whole number part of 3.
• \(3 \dfrac{4}{5}\) has a whole number part of 3 but \(\dfrac{4}{5} > \dfrac{1}{5}\).
• \(4 \dfrac{5}{6}\) has a whole number part of 4.
• \(5 \dfrac{3}{4}\) has a whole number part of 5.

Thus, the correct order from least to greatest is: $$3 \dfrac{1}{5}, \quad 3 \dfrac{4}{5}, \quad 4 \dfrac{5}{6}, \quad 5 \dfrac{3}{4}$$

Correct Answer: B

Since all the mixed numbers have the same whole number part (7), we only need to compare their fractional parts. To make comparison easier, we write all fractions with the lowest common denominator, which is 660.

$$\dfrac{2}{3} = \dfrac{440}{660}, \quad \dfrac{3}{5} = \dfrac{396}{660}, \quad \dfrac{3}{4} = \dfrac{495}{660}, \quad \dfrac{6}{11} = \dfrac{360}{660}$$

Now, compare the numerators:

• \(\dfrac{360}{660}\) is the smallest (from \(7 \dfrac{6}{11}\))
• \(\dfrac{396}{660}\) (from \(7 \dfrac{3}{5}\))
• \(\dfrac{440}{660}\) (from \(7 \dfrac{2}{3}\))
• \(\dfrac{495}{660}\) is the largest (from \(7 \dfrac{3}{4}\))

Therefore, the order from least to greatest is: $$7 \dfrac{6}{11}, \quad 7 \dfrac{3}{5}, \quad 7 \dfrac{2}{3}, \quad 7 \dfrac{3}{4}$$

Correct Answer: C

A fraction represents a part of a whole. Since 1 hour = 60 minutes, we compare 50 minutes to 60 minutes:

$$\text{Fraction of an hour} = \dfrac{50}{60}$$

Now reduce the fraction by dividing numerator and denominator by 10:

$$\dfrac{50}{60} = \dfrac{5}{6}$$

Correct Answer: C

Let the unknown number be \(n\). We can translate the problem into an equation:

$$\dfrac{1}{3} = \dfrac{1}{8} \times n$$

Simplify the right-hand side:

$$\dfrac{1}{3} = \dfrac{n}{8}$$

Now solve for \(n\) by multiplying both sides by 8:

$$n = 8 \times \dfrac{1}{3} = \dfrac{8}{3}$$

Correct Answer: A