# Solutions and Explanations to Questions on Fractions - Grade 5

Solutions and explanations to grade 5 fractions questions are presented.

1. Use any whole number $$n$$ to write 1 as a fraction as follows:
$$n = 1$$, $$\quad 1 = \dfrac{1}{1}$$
$$n = 2$$, $$\quad 1 = \dfrac{2}{2}$$
$$n = 11$$, $$\quad 1 = \dfrac{11}{11}$$
and so on
NOTE that we cannot write $$1 = \dfrac{0}{0}$$.
NOTE that a fraction cannot have a denominator equal to zero.

2. Any whole number $$n$$ may be written as a reduced fraction as follows: $\dfrac{n}{1}$ Hence $$5$$ may be written as $\dfrac{5}{1}$

3. When adding fractions, it is important to have a common denominator. In this case, both fractions have a denominator of 4, so we can add the numerators directly. The sum of the numerators gives us the numerator of the resulting fraction. The denominator remains the same. So, the resulting fraction is $\dfrac{1}{4} + \dfrac{2}{4} = \dfrac{1+2}{4} = \dfrac{3}{4}$

4. When subtracting fractions, it is important to have a common denominator. In this case, both fractions have a denominator of 7, so we can subtract the numerators directly. The difference between the numerators gives us the numerator of the resulting fraction. The denominator remains the same. So, the resulting fraction is $\dfrac{4}{7} - \dfrac{2}{7} = \dfrac{4 - 2}{7} = \dfrac{2}{7}$

5. $\dfrac{1}{5} + \dfrac{2}{3} =$ To add the fractions, we need to follow these steps:
Step 1: Find a common denominator.
In this case, the denominators are different (5 and 3). To find a common denominator, we can multiply the denominators together:
$$5 \times 3 = 15$$
Step 2: Rewrite the fractions so that they have the same denominator. To make the denominators equal to 15, we need to scale the fractions accordingly.
We multiply the numerator and denominator of $$\dfrac{1}{5}$$ by 3, and the numerator and denominator of $$\dfrac{2}{3}$$ by 5:
$$\dfrac{1}{5} = \dfrac{1}{5} \times \dfrac{3}{3} = \dfrac{3}{15}$$
$$\dfrac{2}{3} = \dfrac{2}{3} \times \dfrac{5}{5} = \dfrac{10}{15}$$
Now, both fractions have the same denominator of 15.
$$\dfrac{3}{15} + \dfrac{10}{15} = \dfrac{3+10}{13} = \dfrac{13}{15}$$
Therefore, the sum of
$\dfrac{1}{5} + \dfrac{2}{3} = \dfrac{13}{15}$

6. To add the mixed numbers $$3 \dfrac{1}{2}$$ and $$5 \dfrac{1}{3}$$, we can follow these steps:
Step 1: Analyze the whole parts of $$3 \dfrac{1}{2}$$ and $$5 \dfrac{1}{3}$$.
The whole part of $$3 \dfrac{1}{2}$$ is $$3$$, and the whole part of $$5 \dfrac{1}{3}$$ is $$5$$.
Step 2: Analyze the fractions: The fraction part of $$3 \dfrac{1}{2}$$ is $$\dfrac{1}{2}$$ and the fraction part of $$5 \dfrac{1}{3}$$ is $$\dfrac{1}{3}$$
Step 3: Add the whole parts.
We add the whole parts together: $$3 + 5 = 8$$
Step 4: Add the fraction parts: $$\dfrac{1}{2} + \dfrac{1}{3}$$
Step 5: Find a common denominator.
To add the fractions, we need to find a common denominator. In this case, the least common multiple (LCM) of 2 and 3 is 6.
Step 6: Multiply (adjust) the fractions to have a common denominator of 6:
$$\dfrac{1}{2} = \dfrac{1}{2} \times \dfrac{3}{3} = \dfrac{3}{6}$$
$$\dfrac{1}{3} = \dfrac{1}{3} \times \dfrac{2}{2} = \dfrac{2}{6}$$
$$\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}$$
Step 8: Put all together.
$3 \dfrac{1}{2} + 5 \dfrac{1}{3} = (3+5) + \dfrac{1}{2} + \dfrac{1}{3} = 8 + \dfrac{5}{6}$

7. The total time for Julia to be ready for school is
$$\dfrac{1}{2}\text{ hour} + \dfrac{1}{4} \text{ hour} = ( \dfrac{1}{2} + \dfrac{1}{4} ) \text{ hour}$$
Write fractions with the same denominator
$$\dfrac{1}{2} + \dfrac{1}{4} = \dfrac{1}{2} \times \dfrac{2}{2} + \dfrac{1}{4} = \dfrac{2}{4} + \dfrac{1}{4} = \dfrac{3}{4} \text{ hour}$$.

8. It is easier to compare fractions if they are written with the same denominator
A)
$$\dfrac{5}{2}$$ and $$\dfrac{2}{5}$$ with same denominator become
$$\dfrac{5}{2} = \dfrac{5}{2} \times \dfrac{5}{5} = \dfrac{25}{10}$$
$$\dfrac{2}{5} = \dfrac{2}{5} \times \dfrac{2}{2} = \dfrac{4}{10}$$
Therefore $$\dfrac{5}{2}$$ and $$\dfrac{2}{5}$$ are not equivalent
B)
Write $$\dfrac{4}{3}$$ with denominator 8 as follows
$$\dfrac{4}{3} = \dfrac{4}{3} \times \dfrac{2}{2} = \dfrac{8}{6}$$
Therefore $$\dfrac{4}{3}$$ and $$\dfrac{8}{6}$$ are equivalent
The fractions in parts C) and D) already have the same denominators and are not equivalent.
Conclusion: Fractions 4/3 and 8/6 are equivalent because when written with common denominator both denominators and numerators are equal.

9. To subtract the mixed numbers $$5 \dfrac{2}{3}$$ and $$3 \dfrac{1}{2}$$, we follow these steps:
Step 1: Convert the mixed numbers to improper fractions.
$$5 \dfrac{2}{3} = 5 + \dfrac{2}{3} = 5 \dfrac{3}{3} + \dfrac{2}{3} = \dfrac{15}{3} + \dfrac{2}{3} = \dfrac{17}{3}$$
and
$$3 \dfrac{1}{2} = 3 + \dfrac{1}{2} = 3 \dfrac{2}{2} + \dfrac{1}{2} = \dfrac{6}{2}+\dfrac{1}{2} = \dfrac{7}{2}$$,
Step 2: Find a common denominator to the 2 fractions: The denominators of the fractions are 3 and 2, which are different. To find a common denominator, we multiply them: $$3 \times 2 = 6$$.
Step 2: Write the fractions with a common denominator.
$$\dfrac{17}{3} = \dfrac{17}{3} \times \dfrac{2}{2} = \dfrac{34}{6}$$
$$\dfrac{7}{2} = \dfrac{7}{2} \times \dfrac{3}{3} = \dfrac{21}{6}$$,
Step 4: Subtract the adjusted fractions.
$$5 \dfrac{2}{3} - 3 \dfrac{1}{2} = \dfrac{34}{6} - \dfrac{21}{6} = \dfrac{13}{6}$$
Step 5: Reduce (if possible) and convert the improper fraction back to a mixed number (if desired).
$$\dfrac{13}{6}$$ cannot be reduced but can be written as a mixed number
$$\dfrac{13}{6} = \dfrac{12+1}{6} = \dfrac{12}{6} + \dfrac{1}{6} = 2 \dfrac{1}{6}$$

10. John ate more than Billy and the difference is given by
$$1 \dfrac{2}{3} - 1 \dfrac{1}{4} = (1 - 1) + (\dfrac{2}{3} - \dfrac{1}{4}) = (\dfrac{2}{3} - \dfrac{1}{4})$$
Write fractions with the same denominator
$$\dfrac{2}{3} = \dfrac{2}{3} \times \dfrac{4}{4} = \dfrac{8}{12}$$
$$\dfrac{1}{4}= \dfrac{1}{4} \times \dfrac{3}{3} = \dfrac{3}{12}$$
The difference is
$$1 \dfrac{2}{3} - 1 \dfrac{1}{4} = \dfrac{8}{12} - \dfrac{3}{12} = \dfrac{5}{12}$$
John ate $$\dfrac{5}{12}$$ of a pizza more than Billy.

11. To divide two fractions, you multiply the first one by the multiplicative inverse of the second
The multiplicative inverse of fraction $$\dfrac{a}{b}$$ is the fraction $$\dfrac{b}{a}$$
Change the division of two fractions into a multiplication as follows
$$\dfrac{5}{2} \div \dfrac{3}{4} = \dfrac{5}{2} \times \dfrac{4}{3} = \dfrac{5 \times 4}{2 \times 3} = \dfrac{20}{6}$$
The result is an improper fraction and may be written as a mixed number as follows:
$$\dfrac{20}{6} = \dfrac{18+2}{6} = \dfrac{18}{6} + \dfrac{2}{6} = 3 + \dfrac{2}{6}$$
The fraction $$\dfrac{2}{6}$$ may be reduced by dividing its numerator and denominator by $$2$$
$$\dfrac{2}{6} = \dfrac{2 \div 2}{6 \div 2} = \dfrac{1}{3}$$
Finally
$$\dfrac{5}{2} \div \dfrac{3}{4} = 3 \dfrac{1}{3}$$

12. To divide two fractions, you multiply the first one by the multiplicative inverse of the second
$$5 \div \dfrac{1}{7} = \dfrac{5}{1} \times \dfrac{7}{1} = \dfrac{5 \times 7}{1 \times 1} = \dfrac{35}{1} = 35$$

13. Multiply numerators together and denominators together.
$$\dfrac{2}{5} \times \dfrac{3}{7} = \dfrac{2 \times 3}{5 \times 7} = \dfrac{6}{35}$$

14. Write the given equation
$$a + 1 \dfrac{3}{4} = 2$$
Subtract $$1 \dfrac{3}{4}$$ from both sides of the equation
$$a + 1 \dfrac{3}{4} - 1 \dfrac{3}{4} = 2 - 1 \dfrac{3}{4}$$
Simplify to obtain
$$a = 2 - 1 \dfrac{3}{4}$$
Simplify the right side
$$= 2 - 1 - \dfrac{3}{4}$$
Simplify
$$1 - \dfrac{3}{4}$$
Rewrite $$1$$ as a fraction $$\dfrac{4}{4}$$
$$= \dfrac{4}{4} - \dfrac{3}{4} = \dfrac{1}{4}$$
Hence
$$a = 1/4$$

15. There are two whole shaded items above and one shaded at $$\dfrac{3}{4}$$. Hence the mixed number
$$2 \dfrac{3}{4}$$ represents the shaded parts.

16. False: $$2 \dfrac{1}{2}$$ is a mixed number and is equal to
$$2 \dfrac{1}{2} = 2 + \dfrac{1}{2}$$

17. Let us say she worked $$n$$ hour on Friday. The total (addition) for the 5 days is 15 hours. Let us add all hours for 5 days
$$3 \dfrac{1}{2} + 4 + 2 \dfrac{1}{6} + 1 \dfrac{1}{2} + n = 15$$
Add whole numbers together and fractions together
$$(3 + 4 + 2 + 1) + (\dfrac{1}{2} + \dfrac{1}{6} + \dfrac{1}{2} ) + n = 15$$
Simplify the expressions within the brackets on the left side
$$10 + (1 + \dfrac{1}{6}) + n = 15$$
Which also simplifies to
$$11 + \dfrac{1}{6} + n = 15$$
Subtract $$11 + \dfrac{1}{6}$$ from both sides of the above equation
$$11 + \dfrac{1}{6} + n - 11 - \dfrac{1}{6} = 15 - 11 - \dfrac{1}{6}$$
Simplifies the left and the right sides to obtain
$$n = 4 - \dfrac{1}{6}$$ Rewrite $$4$$ as a fraction with denominator $$4$$
$$n = \dfrac{16}{4} - \dfrac{1}{6}$$
$$n = \dfrac{15}{4}$$
It is an improper fraction that can be written as a mixed number
$$n = \dfrac{15}{4} = \dfrac{12 + 3}{4} = 3 \dfrac{3}{4}$$
Tina worked 3 and $$\dfrac{3}{4}$$ hours on Friday.

18. Write $$1 \dfrac{7}{10}$$ in decimal form as follows
$$1 \dfrac{7}{10} = 1 + 7 \div 10 = 1 + 0.7 = 1.7$$ and corresponds to point W.
$$1.7$$ and corresponds to point W on the graph.

19. Write mixed number as a sum of a whole part and a fractional part
$$2 \dfrac{1}{3} = 2 + \dfrac{1}{3}$$
Write $$2$$ as a fraction with denominator $$3$$
$$= \dfrac{2}{1} \times \dfrac{3}{3} + \dfrac{1}{3}$$
Simplify
$$= \dfrac{6}{3} + \dfrac{1}{3}$$
$$= \dfrac{7}{3}$$
$$2 \dfrac{1}{3}$$ as an
$$2 \dfrac{1}{3} = \dfrac{7}{3}$$

20. Divide 31 by 8 to obtain a quotient equal to 3 and a remainder equal to 7 which can be written as
$$31 = 3 \times 8 + 7$$
Hence we can write that
$$\dfrac{31}{8} = \dfrac{3 \times 8 + 7}{8} = \dfrac{3 \times 8}{8} + \dfrac{7}{8}$$
Simplify
$$= 3 + \dfrac{7}{8} = 3\dfrac{7}{8}$$
$$\dfrac{31}{8}$$ as a mixed number is equal to $$3\dfrac{7}{8}$$

21. $$3 \times \dfrac{1}{4}$$ may be written as
$$3 \times \dfrac{1}{4} = (1 + 1 + 1) \times \dfrac{1}{4}$$
Use distribution
$$=\dfrac{1}{4} + \dfrac{1}{4} + \dfrac{1}{4}$$

22. $$3 \dfrac{1}{4}$$ is a mixed number with a whole part equal to 3 and a fractional part equal to $$\dfrac{1}{4}$$ and is written as
$$3 \dfrac{1}{4} = 3 + \dfrac{1}{4}$$

23. Rewrite the two fractions with the same denominator. The same denominator is the lowest common multiple (LCM) of 5 and 8. First list the first few multiples of 5 and 8 until we obtain a common multiple
factors of 5:   5, 10, 15, 20, 25, 30, 35, 40, ...
factors of 8:   8, 16, 24, 32, 40, ...
The LCM of 5 and 8 is 40
Rewrite the two fractions with the same denominator 40 (which is the LCM)
$$\dfrac{2}{5} = \dfrac{2}{5} \times \dfrac{8}{8} = \dfrac{16}{40}$$
$$\dfrac{3}{8} = \dfrac{3}{8} \times \dfrac{5}{5} = \dfrac{15}{40}$$
$$\dfrac{16}{40}$$ is greater than $$\dfrac{15}{40}$$ and therefore $$\dfrac{2}{5}$$ is greater than $$\dfrac{3}{8}$$ and therefore the above statement is true.

24. The fraction $$\dfrac{7}{6}$$ has its numerator greater than its denominator and hence it is greater than 1.
The remaining 3 fractions $$\dfrac{3}{5} \; , \; \dfrac{1}{3}$$ and $$\dfrac{4}{9}$$ have their numerators smaller than their denominators and are therefore all less than 1. They can be compared by first writing them with the same denominator.
The same denominator may be the lowest common multiple of their denominators 5, 3 and 9.
factors of 5:   5, 10, 15, 25, 30, 35, 40, 45,...
factors of 3:   3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45,...
factors of 9:   9, 18, 27, 36, 45,...
The lowest common multiple of the denominators 5, 3 and 9 is 45. Hence we rewrite the three fractions with the common denominator 45 as follows:
$$\dfrac{3}{5} = \dfrac{3}{5} \times \dfrac{9}{9} = \dfrac{27}{45}$$
$$\dfrac{1}{3} = \dfrac{1}{3} \times \dfrac{15}{15} = \dfrac{15}{45}$$
$$\dfrac{4}{9} = \dfrac{4}{9} \times \dfrac{5}{5} = \dfrac{20}{45}$$
Using the above fractions, we now order the given fractions from least to greatest as follows
$$\dfrac{1}{3} \; , \; \dfrac{4}{9} \; , \; \dfrac{3}{5} \; , \; \dfrac{7}{6}$$

25. $$\dfrac{2}{3}$$ of $$4$$ is equal to:
$$\dfrac{2}{3} \times 4 = \dfrac{2}{3} \times \dfrac{4}{1} = \dfrac{2 \times 4}{3 \times 1}$$
Simplify
$$= \dfrac{8}{3}$$
Write 8 as 6 + 2. (6 is a multiple of 3)
$$= \dfrac{6 + 2 }{3}$$
Rewrite as a sum of fractions
$$= \dfrac{6}{3} + \dfrac{2 }{3}$$
Simplify
$$= 2 + \dfrac{2 }{3} = 2 \dfrac{2 }{3}$$
Hence $$\dfrac{2}{3}$$ of $$4$$ as a mixed number is equal to
$$2 \dfrac{2 }{3}$$

26. One hour is equal to 60 minutes. Hence $$\dfrac{2}{3}$$ of an hour is equal to
$$\dfrac{2}{3} \times 60$$
Rewrite 60 as a fraction $$\dfrac{60}{1}$$
$$= \dfrac{2}{3} \times \dfrac{60}{1}$$
Multiply fractions and simplify
$$= \dfrac{2 \times 60}{3 \times 1} = \dfrac{120}{3}$$
Rewrite fraction as a division and simplify
$$= 120 \div 3 = 40$$ minutes
Conclusion: Hence $$\dfrac{2}{3}$$ of an hour is equal to 40 minutes.

27. The large square is divided into 16 small squares. Hence every small square is $$\dfrac{1}{16}$$ of the large square.
red: 4 small squares represent $$4 \times \dfrac{1}{16} = \dfrac{4}{16} = \dfrac{1}{4}$$ of the large square
blue: 1 small square represents $$1 \times \dfrac{1}{16} = \dfrac{1}{16}$$ of the large square
orange: half a small square represents $$\dfrac{1}{2}$$ of $$\dfrac{1}{16}$$ = $$\dfrac{1}{2} \times \dfrac{1}{16} = \dfrac{1}{32}$$ of the large square
green: 1 small square and 1/2 of a small square represents $$\dfrac{1}{16} + \dfrac{1}{2} \times \dfrac{1}{16}$$
Simplify
$$= \dfrac{1}{16} + \dfrac{1}{32}$$
Rewrite the fraction $$\dfrac{1}{16}$$ with denominator 32
$$= \dfrac{1}{16} \times \dfrac{2}{2} + \dfrac{1}{32}$$
Simplify
$$= \dfrac{3}{32}$$ of the large square
black: 3 small squares represent $$3 \times \dfrac{1}{16} = \dfrac{3}{16}$$ of the large square
yellow: 3 small squares represent $$3 \times \dfrac{1}{16} = \dfrac{3}{16}$$ of the large square
We can write the color with the corresponding fractions as follows:
red: $$\dfrac{1}{4}$$ , blue: $$\dfrac{1}{16}$$ , orange: $$\dfrac{1}{32}$$ , green: $$\dfrac{3}{32}$$ , black: $$\dfrac{3}{16}$$ , yellow: $$\dfrac{3}{16}$$