Fractions | FIO

Question 31

Write the following fractions in ascending order.

a. $\dfrac{7}{10}$ , $\dfrac{11}{15}$ , $\dfrac{2}{5}$

b. $\dfrac{19}{24}$ , $\dfrac{5}{6}$ , $\dfrac{7}{12}$

Solution

Understand the Question

To arrange fractions in ascending order (from smallest to largest), they must first be converted into like fractions having a common denominator.
The common denominator is found by taking the Least Common Multiple (LCM) of the denominators.
Once all fractions share the same denominator, we compare their numerators: the fraction with the smaller numerator is smaller in value.

a. $\dfrac{7}{10}$ , $\dfrac{11}{15}$ , $\dfrac{2}{5}$

Step 1 · Find the LCM of Denominators

The denominators are $10$ , $15$ , and $5$ .

Listing multiples:

Multiples of $10$ : $10, 20, 30, 40, \dots$
Multiples of $15$ : $15, 30, 45, \dots$
Multiples of $5$ : $5, 10, 15, 20, 25, 30, \dots$

The smallest common multiple is $30$ .

$\text{LCM}(10, 15, 5) = 30$

Step 2 · Convert Fractions to Like Fractions

Convert each fraction to an equivalent fraction with denominator $30$ :

\begin{aligned} \dfrac{7}{10} &= \dfrac{7 \times 3}{10 \times 3} = \dfrac{21}{30} \\[0.6em] \dfrac{11}{15} &= \dfrac{11 \times 2}{15 \times 2} = \dfrac{22}{30} \\[0.6em] \dfrac{2}{5} &= \dfrac{2 \times 6}{5 \times 6} = \dfrac{12}{30} \end{aligned}

Step 3 · Compare Numerators and Arrange in Ascending Order

Comparing the numerators: $12 < 21 < 22$

Therefore, $\dfrac{12}{30} < \dfrac{21}{30} < \dfrac{22}{30}$

$\dfrac{2}{5} < \dfrac{7}{10} < \dfrac{11}{15}$

Answer

a. $\dfrac{2}{5} < \dfrac{7}{10} < \dfrac{11}{15}$

b. $\dfrac{19}{24}$ , $\dfrac{5}{6}$ , $\dfrac{7}{12}$

Step 1 · Find the LCM of Denominators

The denominators are $24$ , $6$ , and $12$ .

Listing multiples:

Multiples of $24$ : $24, 48, \dots$
Multiples of $6$ : $6, 12, 18, 24, 30, \dots$
Multiples of $12$ : $12, 24, 36, \dots$

The smallest common multiple is $24$ .

$\text{LCM}(24, 6, 12) = 24$

Step 2 · Convert Fractions to Like Fractions

Convert each fraction to an equivalent fraction with denominator $24$ :

\begin{aligned} \dfrac{19}{24} &= \dfrac{19}{24} \\[0.6em] \dfrac{5}{6} &= \dfrac{5 \times 4}{6 \times 4} = \dfrac{20}{24} \\[0.6em] \dfrac{7}{12} &= \dfrac{7 \times 2}{12 \times 2} = \dfrac{14}{24} \end{aligned}

Step 3 · Compare Numerators and Arrange in Ascending Order

Comparing the numerators: $14 < 19 < 20$

Therefore, $\dfrac{14}{24} < \dfrac{19}{24} < \dfrac{20}{24}$

$\dfrac{7}{12} < \dfrac{19}{24} < \dfrac{5}{6}$

Answer

b. $\dfrac{7}{12} < \dfrac{19}{24} < \dfrac{5}{6}$

Common Mistakes

Comparing Numerators Directly: Comparing numerators directly without converting unlike fractions to like fractions leads to incorrect ordering.
Ascending vs. Descending Order: Confusing ascending order (smallest to largest) with descending order (largest to smallest).
Unequal Multiplication: Forgetting to multiply both the numerator and the denominator by the same factor when forming equivalent fractions.

More questions in FIO

Fill in the blanks with fractions.

Three guavas together weigh 1 kg. If they are roughly of the same size, each guava will roughly weigh ___ kg.

A wholesale merchant packed $1\text{ kg}$ of rice in four packets of equal weight. The weight of each packet is $\underline{\hspace{2em}}\text{ kg}$ .

Four friends ordered 3 glasses of sugarcane juice and shared it equally among themselves. Each one drank ___ glass of sugarcane juice.

The big fish weighs $\dfrac{1}{2}$ kg. The small one weighs $\dfrac{1}{4}$ kg. Together they weigh ___ kg.

The figures below show different fractional units of a whole chikki. How much of a whole chikki is each piece?

Continue this table of $\dfrac{1}{2}$ for 2 more steps.

Can you create a similar table for $\dfrac{1}{4}$ ?

Make $\dfrac{1}{3}$ using a paper strip. Can you use this to also make $\dfrac{1}{6}$ ?

Draw a picture and write an addition statement as above to show:

a. 5 times $\dfrac{1}{4}$ of a roti

b. 9 times $\dfrac{1}{4}$ of a roti

Q10

Match each fractional unit with the correct picture:

Q11

On a number line, draw lines of lengths $\dfrac{1}{10}$ , $\dfrac{3}{10}$ , and $\dfrac{4}{5}$ .

Q12

Write five more fractions of your choice and mark them on the number line.

Q13

How many fractions lie between $0$ and $1$ ? Think, discuss with your classmates, and write your answer.

Q14

What is the length of the blue line and black line shown below? The distance between 0 and 1 is 1 unit long, and it is divided into two equal parts. The length of each part is $\dfrac{1}{2}$ . So the blue line is $\dfrac{1}{2}$ units long. Write the fraction that gives the length of the black line in the box.

Q15

Write the fraction that gives the lengths of the black lines in the respective boxes.

Q16

How many whole units are there in $\dfrac{7}{2}$ ?

Q17

How many whole units are there in $\dfrac{4}{3}$ and in $\dfrac{7}{3}$ ?

Q18

Figure out the number of whole units in each of the following fractions:

a. $\dfrac{8}{3}$

b. $\dfrac{11}{5}$

c. $\dfrac{9}{4}$

Q19

Can all fractions greater than $1$ be written as such mixed numbers?

Q20

Write the following fractions as mixed fractions (e.g., $\dfrac{9}{2} = 4\dfrac{1}{2}$ ):

a. $\dfrac{9}{2}$

b. $\dfrac{9}{5}$

c. $\dfrac{21}{19}$

d. $\dfrac{47}{9}$

e. $\dfrac{12}{11}$

f. $\dfrac{19}{6}$

Q21

Write the following mixed numbers as fractions:

a. $3 \dfrac{1}{4}$

b. $7 \dfrac{2}{3}$

c. $9 \dfrac{4}{9}$

d. $3 \dfrac{1}{6}$

e. $2 \dfrac{3}{11}$

f. $3 \dfrac{9}{10}$

Q22

Are $\dfrac{3}{6}$ , $\dfrac{4}{8}$ , $\dfrac{5}{10}$ equivalent fractions? Why?

Q23

Write two equivalent fractions for $\dfrac{2}{6}$ .

Q24

$\dfrac{4}{6} = \dots\dots\dots\dots\dots$ (Write as many as you can)

Q25

Three rotis are shared equally by four children. Show the division in the picture and write a fraction for how much each child gets. Also, write the corresponding division facts, addition facts, and, multiplication facts.

Fraction of roti each child gets is ______. Division fact: Addition fact: Multiplication fact:

Compare your picture and answers with your classmates!

Q26

Draw a picture to show how much each child gets when 2 rotis are shared equally by 4 children. Also, write the corresponding division facts, addition facts, and multiplication facts.

Q27

Anil was in a group where 2 cakes were divided equally among 5 children. How much cake would Anil get?

Q28

Find the missing numbers:

a. 5 glasses of juice shared equally among 4 friends is the same as ___ glasses of juice shared equally among 8 friends.

b. 4 kg of potatoes divided equally in 3 bags is the same as 12 kgs of potatoes divided equally in ___ bags.

c. 7 rotis divided among 5 children is the same as ___ rotis divided among ___ children.

Q29

Express the following fractions in lowest terms:

a. $\dfrac{17}{51}$

b. $\dfrac{64}{144}$

c. $\dfrac{126}{147}$

d. $\dfrac{525}{112}$

Q30

Compare the following fractions and justify your answers:

a. $\dfrac{8}{3}$ , $\dfrac{5}{2}$

b. $\dfrac{4}{9}$ , $\dfrac{3}{7}$

c. $\dfrac{7}{10}$ , $\dfrac{9}{14}$

d. $\dfrac{12}{5}$ , $\dfrac{8}{5}$

e. $\dfrac{9}{4}$ , $\dfrac{5}{2}$

Q31

Write the following fractions in ascending order.

a. $\dfrac{7}{10}$ , $\dfrac{11}{15}$ , $\dfrac{2}{5}$

b. $\dfrac{19}{24}$ , $\dfrac{5}{6}$ , $\dfrac{7}{12}$

Q32

Write the following fractions in descending order.

a. $\dfrac{25}{16}$ , $\dfrac{7}{8}$ , $\dfrac{13}{4}$ , $\dfrac{17}{32}$

b. $\dfrac{3}{4}$ , $\dfrac{12}{5}$ , $\dfrac{7}{12}$ , $\dfrac{5}{4}$

Q33

Add the following fractions using Brahmagupta's method:

a. $\dfrac{2}{7} + \dfrac{5}{7} + \dfrac{6}{7}$

b. $\dfrac{3}{4} + \dfrac{1}{3}$

c. $\dfrac{2}{3} + \dfrac{5}{6}$

d. $\dfrac{2}{3} + \dfrac{2}{7}$

e. $\dfrac{3}{4} + \dfrac{1}{3} + \dfrac{1}{5}$

f. $\dfrac{2}{3} + \dfrac{4}{5}$

g. $\dfrac{4}{5} + \dfrac{2}{3}$

h. $\dfrac{3}{5} + \dfrac{5}{8}$

i. $\dfrac{9}{2} + \dfrac{5}{4}$

j. $\dfrac{8}{3} + \dfrac{2}{7}$

k. $\dfrac{3}{4} + \dfrac{1}{3} + \dfrac{1}{5}$

l. $\dfrac{2}{3} + \dfrac{4}{5} + \dfrac{3}{7}$

m. $\dfrac{9}{2} + \dfrac{5}{4} + \dfrac{7}{6}$

Q34

Rahim mixes $\dfrac{2}{3}$ litres of yellow paint with $\dfrac{3}{4}$ litres of blue paint to make green paint. What is the volume of green paint he has made?

Q35

Geeta bought $\dfrac{2}{5}$ meter of lace and Shamim bought $\dfrac{3}{4}$ meter of the same lace to put a complete border on a table cloth whose perimeter is $1$ meter long. Find the total length of the lace they both have bought. Will the lace be sufficient to cover the whole border?

Q36

$\dfrac{5}{8} - \dfrac{3}{8}$

Q37

$\dfrac{7}{9} - \dfrac{5}{9}$

Q38

$\dfrac{10}{27} - \dfrac{1}{27}$

Q39

Carry out the following subtractions using Brahmagupta's method:

a. $\dfrac{8}{15} - \dfrac{3}{15}$

b. $\dfrac{2}{5} - \dfrac{4}{15}$

c. $\dfrac{5}{6} - \dfrac{4}{9}$

d. $\dfrac{2}{3} - \dfrac{1}{2}$

Q40

Subtract as indicated:

a. $\dfrac{13}{4}$ from $\dfrac{10}{3}$

b. $\dfrac{18}{5}$ from $\dfrac{23}{3}$

c. $\dfrac{29}{7}$ from $\dfrac{45}{7}$

Q41

Solve the following problems:

a. Jaya's school is $\dfrac{7}{10}$ km from her home. She takes an auto for $\dfrac{1}{2}$ km from her home daily, and then walks the remaining distance to reach her school. How much does she walk daily to reach the school?

b. Jeevika takes $\dfrac{10}{3}$ minutes to take a complete round of the park and her friend Namit takes $\dfrac{13}{4}$ minutes to do the same. Who takes less time and by how much?