Working with Fractions | FIO

Question 24

A colony of ants set out in search of food. As they search, they keep splitting equally at each point (as shown in the Fig. 8.7) and reach two food sources, one near a mango tree and another near a sugarcane field. What fraction of the original group reached each food source?

Solution

Understand the Question

The entire colony of ants starts as $1$ whole unit at the beginning of their journey.
At each junction (red circle), the incoming fraction of ants divides equally among the outgoing paths.
Branches terminate at either the Mango tree or the Sugarcane field, while others continue to the next junction.
The total fraction of ants reaching each food source is the sum of fractions along all paths leading to that destination.

Step 1 · First Split

Let the total group of ants be $1$ . Diagram 1

At the first junction, the ants split equally into two paths: $\text{Fraction on left path} = \dfrac{1}{2}$

$\text{Fraction on right path} = \dfrac{1}{2}$

Step 2 · First Path to Mango Tree

The left path from the first split goes directly to the Mango tree. Diagram 2

$\text{Ants to Mango tree (Path A)} = \dfrac{1}{2}$

Step 3 · Second Split

The right path carrying $\dfrac{1}{2}$ of the ants reaches the second junction and splits into two equal paths. Diagram 3

\begin{aligned} \text{Fraction on left sub-path} &= \dfrac{1}{2} \times \dfrac{1}{2} \\[0.6em] &= \dfrac{1}{4} \end{aligned}

\begin{aligned} \text{Fraction on right sub-path} &= \dfrac{1}{2} \times \dfrac{1}{2} \\[0.6em] &= \dfrac{1}{4} \end{aligned}

Step 4 · Second Path to Mango Tree

The left sub-path from the second split leads directly to the Mango tree. Diagram 4

$\text{Ants to Mango tree (Path B)} = \dfrac{1}{4}$

Step 5 · Third Split

The right sub-path carrying $\dfrac{1}{4}$ of the ants reaches the third junction and splits into $4$ equal paths. Diagram 5

\begin{aligned} \text{Fraction on each of the four new paths} &= \dfrac{1}{4} \times \dfrac{1}{4} \\[0.6em] &= \dfrac{1}{16} \end{aligned}

Step 6 · Paths from Third Split

Two of the four paths go directly to the Mango tree, one path goes to the Sugarcane field, and one continues to the next split. Diagram 6

\begin{aligned} \text{Ants to Mango tree (Path C and D)} &= \dfrac{1}{16} + \dfrac{1}{16} \\[0.6em] &= \dfrac{2}{16} \end{aligned}

$\text{Ants to Sugarcane field (Path E)} = \dfrac{1}{16}$

Step 7 · Fourth Split

The remaining path carrying $\dfrac{1}{16}$ of the ants reaches the fourth junction and splits into $2$ equal paths. Diagram 7

\begin{aligned} \text{Fraction on each of the two new paths} &= \dfrac{1}{16} \times \dfrac{1}{2} \\[0.6em] &= \dfrac{1}{32} \end{aligned}

Step 8 · Final Paths to Food Sources

One path goes to the Mango tree and the other goes to the Sugarcane field. Diagram 8

$\text{Ants to Mango tree (Path F)} = \dfrac{1}{32}$

$\text{Ants to Sugarcane field (Path G)} = \dfrac{1}{32}$

Step 9 · Total Fraction Reaching Mango Tree

Sum the fractions of ants from Paths A, B, C, D, and F: Diagram 9

\begin{aligned} \text{Total ants to Mango tree} &= \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{2}{16} + \dfrac{1}{32} \\[0.6em] &= \dfrac{1 \times 16}{2 \times 16} + \dfrac{1 \times 8}{4 \times 8} + \dfrac{2 \times 2}{16 \times 2} + \dfrac{1}{32} \\[0.6em] &= \dfrac{16}{32} + \dfrac{8}{32} + \dfrac{4}{32} + \dfrac{1}{32} \\[0.6em] &= \dfrac{16 + 8 + 4 + 1}{32} \\[0.6em] &= \dfrac{29}{32} \end{aligned}

Step 10 · Total Fraction Reaching Sugarcane Field

Sum the fractions of ants from Paths E and G: Diagram 10

\begin{aligned} \text{Total ants to Sugarcane field} &= \dfrac{1}{16} + \dfrac{1}{32} \\[0.6em] &= \dfrac{1 \times 2}{16 \times 2} + \dfrac{1}{32} \\[0.6em] &= \dfrac{2}{32} + \dfrac{1}{32} \\[0.6em] &= \dfrac{2 + 1}{32} \\[0.6em] &= \dfrac{3}{32} \end{aligned}

Answer

Mango tree: $\dfrac{29}{32}$ , Sugarcane field: $\dfrac{3}{32}$

Common Mistakes

Splitting the Original Whole Instead of Current Branch: When a path divides, multiply the fraction entering that junction, not the original total ( $1$ ).
Omitting Parallel Paths: At the 3rd junction, $2$ separate branches lead to the mango tree, contributing $\dfrac{1}{16} + \dfrac{1}{16} = \dfrac{2}{16}$ .
Sanity Check: The sum of all final fractions must equal $1$ : $\dfrac{29}{32} + \dfrac{3}{32} = \dfrac{32}{32} = 1$

More questions in FIO

Tenzin drinks $\dfrac{1}{2}$ glass of milk every day. How many glasses of milk does he drink in a week? How many glasses of milk did he drink in the month of January?

A team of workers can make 1 km of a water canal in 8 days. So, in one day, the team can make ___ km of the water canal. If they work 5 days a week, they can make ___ km of the water canal in a week.

Manju and two of her neighbours buy 5 litres of oil every week and share it equally among the 3 families. How much oil does each family get in a week? How much oil will one family get in 4 weeks?

Safia saw the Moon setting on Monday at 10 pm. Her mother, who is a scientist, told her that every day the Moon sets $\dfrac{5}{6}$ hour later than the previous day. How many hours after 10 pm will the moon set on Thursday?

Multiply and then convert it into a mixed fraction:

(a) $7 \times \dfrac{3}{5}$

(b) $4 \times \dfrac{1}{3}$

(d) $\dfrac{13}{11} \times 6$

Find the following products. Use a unit square as a whole for representing the fractions:

(a) $\dfrac{1}{3} \times \dfrac{1}{5}$

(b) $\dfrac{1}{4} \times \dfrac{1}{3}$

(d) $\dfrac{1}{6} \times \dfrac{1}{5}$

Now, find $\dfrac{1}{12} \times \dfrac{1}{18}$ .

Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations.

(a) $\dfrac{2}{3} \times \dfrac{4}{5}$

(b) $\dfrac{1}{4} \times \dfrac{2}{3}$

(d) $\dfrac{4}{6} \times \dfrac{3}{5}$

A water tank is filled from a tap. If the tap is open for 1 hour, $\dfrac{7}{10}$ of the tank gets filled. How much of the tank is filled if the tap is open for

(a) $\dfrac{1}{3}$ hour

(b) $\dfrac{2}{3}$ hour

(d) $\dfrac{7}{10}$ hour

(e) For the tank to be full, how long should the tap be running?

The government has taken $\dfrac{1}{6}$ of Somu's land to build a road. What part of the land remains with Somu now? She gives half of the remaining part of the land to her daughter Krishna and $\dfrac{1}{3}$ of it to her son Bora. After giving them their shares, she keeps the remaining land for herself.

(a) What part of the original land did Krishna get?

(b) What part of the original land did Bora get?

Q10

Find the area of a rectangle of sides $3\dfrac{3}{4}\text{ ft}$ and $9\dfrac{3}{5}\text{ ft}$ .

Q11

Tsewang plants four saplings in a row in his garden. The distance between two saplings is $\dfrac{3}{4} \text{ m}$ . Find the distance between the first and last sapling. [Hint: Draw a rough diagram with four saplings with distance between two saplings as $\dfrac{3}{4} \text{ m}$ ]

Q12

Which is heavier: $\dfrac{12}{15}$ of 500 grams or $\dfrac{3}{20}$ of 4 kg?

Q13

What can you conclude about the relationship between the numbers multiplied and the product? Fill in the blanks:

When one of the numbers being multiplied is between $0$ and $1$ , the product is ________ (greater/less) than the other number.
When one of the numbers being multiplied is greater than $1$ , the product is ________ (greater/less) than the other number.

Q14

Evaluate the following:

(a) $3 \div \dfrac{7}{9}$

(b) $\dfrac{14}{4} \div 2$

(d) $\dfrac{14}{6} \div \dfrac{7}{3}$

(e) $\dfrac{4}{3} \div \dfrac{3}{4}$

(f) $\dfrac{7}{4} \div \dfrac{1}{7}$

(g) $\dfrac{8}{2} \div \dfrac{4}{15}$

(h) $\dfrac{1}{5} \div \dfrac{1}{9}$

(i) $\dfrac{1}{6} \div \dfrac{11}{12}$

(j) $3\dfrac{2}{3} \div 1\dfrac{3}{8}$

Q15

For each of the questions below, choose the expression that describes the solution. Then simplify it.

(a) Maria bought $8 \text{ m}$ of lace to decorate the bags she made for school. She used $\dfrac{1}{4} \text{ m}$ for each bag and finished the lace. How many bags did she decorate?

(i) $8 \times \dfrac{1}{4}$

(ii) $\dfrac{1}{8} \times \dfrac{1}{4}$

(iii) $8 \div \dfrac{1}{4}$

(iv) $\dfrac{1}{4} \div 8$

(b) $\dfrac{1}{2} \text{ meter}$ of ribbon is used to make 8 badges. What is the length of the ribbon used for each badge?

(i) $8 \times \dfrac{1}{2}$

(ii) $\dfrac{1}{2} \cdot \dfrac{1}{8}$

(iii) $8 \div \dfrac{1}{2}$

(iv) $\dfrac{1}{2} \div 8$

(c) A baker needs $\dfrac{1}{6} \text{ kg}$ of flour to make one loaf of bread. He has $5 \text{ kg}$ of flour. How many loaves of bread can he make?

(i) $5 \times \dfrac{1}{6}$

(ii) $\dfrac{1}{6} \div 5$

(iii) $5 \div \dfrac{1}{6}$

(iv) $5 \times 6$

Q16

If $\dfrac{1}{4}$ kg of flour is used to make 12 rotis, how much flour is used to make 6 rotis?

Q17

Pāṭīgaṇita, a book written by Sridharacharya in the 9th century CE, mentions this problem: "Friend, after thinking, what sum will be obtained by adding together $1 \div \dfrac{1}{6}$ , $1 \div \dfrac{1}{10}$ , $1 \div \dfrac{1}{13}$ , $1 \div \dfrac{1}{9}$ , and $1 \div \dfrac{1}{2}$ ". What should the friend say?

Q18

Mira is reading a novel that has 400 pages. She read $\dfrac{1}{5}$ of the pages yesterday and $\dfrac{3}{10}$ of the pages today. How many more pages does she need to read to finish the novel?

Q19

A car runs 16 km using 1 litre of petrol. How far will it go using $2 \dfrac{3}{4}$ litres of petrol?

Q20

Amritpal decides on a destination for his vacation. If he takes a train, it will take him $5\dfrac{1}{6}$ hours to get there. If he takes a plane, it will take him $\dfrac{1}{2}$ hour. How many hours does the plane save?

Q21

Mariam's grandmother baked a cake. Mariam and her cousins finished $\dfrac{4}{5}$ of the cake. The remaining cake was shared equally by Mariam's three friends. How much of the cake did each friend get?

Q22

Choose the option(s) describing the product of $\left(\dfrac{565}{465} \times \dfrac{707}{676}\right)$ :

(a) $> \dfrac{565}{465}$

(b) $< \dfrac{565}{465}$

(d) $< \dfrac{707}{676}$

(e) $> 1$

(f) $< 1$

Q23

What fraction of the whole square is shaded?

Q24

Q25

What is $1 - \dfrac{1}{2}$ ?

$\left(1 - \dfrac{1}{2}\right) \times \left(1 - \dfrac{1}{3}\right)$ ?

$\left(1 - \dfrac{1}{2}\right) \times \left(1 - \dfrac{1}{3}\right) \times \left(1 - \dfrac{1}{4}\right) \times \left(1 - \dfrac{1}{5}\right)$ ?

$\left(1 - \dfrac{1}{2}\right) \times \left(1 - \dfrac{1}{3}\right) \times \left(1 - \dfrac{1}{4}\right) \times \left(1 - \dfrac{1}{5}\right) \times \left(1 - \dfrac{1}{6}\right) \times \left(1 - \dfrac{1}{7}\right) \times \left(1 - \dfrac{1}{8}\right) \times \left(1 - \dfrac{1}{9}\right) \times \left(1 - \dfrac{1}{10}\right)$ ?

Make a general statement and explain.