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

We will follow the ants from the start, calculating the fraction of the group on each path as they split.

Step 1 — Starting the journey

Let the total group of ants be 1. The ants start their journey. They reach the first red circle. At this point, they split into two equal paths. So, each path gets half of the ants.

$\text{Fraction on left path} = \frac{1}{2}$

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

Diagram 1

Step 2 — Following the left main path

The left path from the first split goes directly to the Mango tree. So, this fraction of ants reaches the Mango tree.

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

Diagram 2

Step 3 — Following the right main path

The right path from the first split continues. It reaches the second red circle. At this point, the ants split into two equal paths again. Each new path gets half of the ants from the previous path.

$\text{Fraction on left sub-path} = \frac{1}{2} \times \frac{1}{2}$

$= \frac{1}{4}$

$\text{Fraction on right sub-path} = \frac{1}{2} \times \frac{1}{2}$

$= \frac{1}{4}$

Diagram 3

Step 4 — Following the left sub-path

The left sub-path from the second split goes directly to the Mango tree. So, this fraction of ants reaches the Mango tree.

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

Diagram 4

Step 5 — Following the right sub-path

The right sub-path from the second split continues. It reaches the third red circle. At this point, the ants split into four equal paths. Each new path gets one-fourth of the ants from the previous path.

$\text{Fraction on each of the four new paths} = \frac{1}{4} \times \frac{1}{4}$

$= \frac{1}{16}$

Diagram 5

Step 6 — Paths from the third split

Two of these four paths go directly to the Mango tree. So, we add these fractions for the Mango tree.

$\text{Ants to Mango tree (Path C and D)} = \frac{1}{16} + \frac{1}{16}$

$= \frac{2}{16}$

One path goes directly to the Sugarcane field. So, this fraction of ants reaches the Sugarcane field.

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

The remaining path continues to another split.

Diagram 6

Step 7 — Following the last splitting path

The remaining path from the third split reaches the fourth red circle. This path had $1/16$ of the ants. At this point, the ants split into two equal paths. Each new path gets half of the ants from the previous path.

$\text{Fraction on each of the two new paths} = \frac{1}{16} \times \frac{1}{2}$

$= \frac{1}{32}$

Diagram 7

Step 8 — Final paths to food sources

One of these two paths goes to the Mango tree. The other path goes to the Sugarcane field.

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

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

Diagram 8

Step 9 — Total ants for Mango tree

We add up all the fractions of ants that reached the Mango tree. These are from Path A, Path B, Path C, Path D, and Path F.

$\text{Total ants to Mango tree} = \frac{1}{2} + \frac{1}{4} + \frac{2}{16} + \frac{1}{32}$

We need a common denominator, which is 32.

$= \frac{1 \times 16}{2 \times 16} + \frac{1 \times 8}{4 \times 8} + \frac{2 \times 2}{16 \times 2} + \frac{1}{32}$

$= \frac{16}{32} + \frac{8}{32} + \frac{4}{32} + \frac{1}{32}$

$= \frac{16 + 8 + 4 + 1}{32}$

$\boxed{\frac{29}{32}}$

Diagram 9

Step 10 — Total ants for Sugarcane field

We add up all the fractions of ants that reached the Sugarcane field. These are from Path E and Path G.

$\text{Total ants to Sugarcane field} = \frac{1}{16} + \frac{1}{32}$

We need a common denominator, which is 32.

$= \frac{1 \times 2}{16 \times 2} + \frac{1}{32}$

$= \frac{2}{32} + \frac{1}{32}$

$= \frac{2 + 1}{32}$

$\boxed{\frac{3}{32}}$

Diagram 10

Answer

(i) The fraction of the original group that reached the mango tree is 29/32. (ii) The fraction of the original group that reached the sugarcane field is 3/32.

More questions in FIO

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Safia saw the Moon setting on Monday at 10 pm. Her mother, who is a scientist, told her that every day the Moon sets $\frac{5}{6}$ 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) $7 \times \frac{3}{5}$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The government has taken $\frac{1}{6}$ 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 $\frac{1}{3}$ 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 $3\frac{3}{4}\text{ ft}$ and $9\frac{3}{5}\text{ ft}$ .

Q11

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

Q12

Which is heavier: $\frac{12}{15}$ of 500 grams or $\frac{3}{20}$ 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) $3 \div \frac{7}{9}$

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

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

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

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

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

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

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

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

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 $\frac{1}{4}\text{ m}$ for each bag and finished the lace. How many bags did she decorate?

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

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

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

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

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

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

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

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

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

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

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

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

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

(iv) $5 \times 6$

Q16

If $\frac{1}{4}$ 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 $1 \div \frac{1}{6}$ , $1 \div \frac{1}{10}$ , $1 \div \frac{1}{13}$ , $1 \div \frac{1}{9}$ , and $1 \div \frac{1}{2}$ ". What should the friend say?

Q18

Mira is reading a novel that has 400 pages. She read $\frac{1}{5}$ of the pages yesterday and $\frac{3}{10}$ 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 $2 \frac{3}{4}$ litres of petrol?

Q20

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

Q21

Mariam's grandmother baked a cake. Mariam and her cousins finished $\frac{4}{5}$ 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 $\left(\frac{565}{465} \times \frac{707}{676}\right)$ :

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

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

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

(e) $> 1$

(f) $< 1$

Q23

What fraction of the whole square is shaded?

Q24

Q25

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

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

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

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

Make a general statement and explain.