Unitary Method, Percentages and Ratio (Class- 6, Experience- 5)

[solve the problems below according to the rules of Ratios, Page 133]

E.5.1. Find the ratios of the first quantity to the second quantity of the following pairs of numbers: 

(a) 25 and 335 

(b) 713  and  925 

(c) 1.25 and 7.5 

(d) 823  and 0.125

(e) 1 year 2 months and 7 months 

(f) 7 kg and 2 kg 300 g 

(g) Tk 2 and 40 poysa

Solution:

let's solve each pair step by step:

(a) 25 and 335

To find the ratio of the first quantity to the second quantity, we simply divide the first quantity by the second quantity.

Ratio =  25335 

          =  567   [divided by 5]

          =5 : 67

(b) 713  and  925 

 OR  22 3 and 47 5

Ratio = 22 3 ÷ 47 5

          = 22 3 ✕  547

          = 22 ✕ 53 ✕ 47

          = 110141

          = 110 : 141

(c) 1.25 and 7.5

Ratio = 1.25 7.5 

          = 125750 [multiply by 100]

          =  530  [divide by 25]

          = 16   [divide by 5]

          = 1 : 6 

(d)  823  and 0.125

Or, 26 3 and 0.125 

To find the ratio of  26 3 to 0.125, you divide 26 3 by 0.125.

Ratio = 26 3 ÷ 0.125

Ratio = 26 3 ✕    10.125  [To divide by a fraction, you multiply by its reciprocal]

Ratio = 26 3 ✕ 1000 125  [multyply by 1000 in the numerators and the denominators]

          = 26 3 ✕ 8

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

Ratio = 26 ✕ 8    3 

          = 208  3 

          = 208 : 3

(e) 1 year 2 months and 7 months

Convert both time intervals to a common unit, such as months.

1 year 2 months = 14 months

Ratio = 14 7 months

          = 21 

          = 2 : 1

(f) 7 kg and 2 kg 300 g

Convert both weights to a common unit, such as grams.

7 kg = 7000 g

2 kg 300 g = 2300 g

Ratio = 70002300 g

          = 7023 g

          = 70 : 23

(g) Tk 2 and 40 poysa

Or, Ratio = 200 40 poysa 

Ratio = 200 40    [Both numbers can be divided by 40]

          = 51

          = 5 : 1

E.5.2. Count the number of books and exercise books you brought to class and complete the following tasks: 

(a) Find the ratio of numbers of exercise books and. 

(b) Find the ratio of the total numbers of pages of the exercise books and of the books.

Solution:

Let Say: 

Number of books = 8 

Number of exercise books = 12 

Pages in each book = 230 

Average Pages in each exercise book = 200

(a) Ratio of numbers of exercise books to books:

Ratio = Number of exercise books     Number of books 

Ratio = 12 8

Ratio = 32    [divided by 4]

Ratio = 3 : 2

So, the ratio of exercise books to books is 3:2.

(b) Ratio of the total numbers of pages of the exercise books and books: 

Total pages in exercise books = 12×200=2400 pages 

Total pages in books = 8×230=1840 pages 

 

Ratio = Total page in exercise books     Total page in books 

Ratio = 24001840

Ratio = 3023    [divided by 80]

Ratio = 30 : 23

So, the ratio of the total numbers of pages of the exercise books to the books is 30 : 23

E.5.3. Find the length and breadth of your Mathematics book using a ruler and find their ratio.

Solution:

Answer will be posted soon. Please wait...

E.5.4. Find 3 different tables in your classroom or home or somewhere else. 

(a) Find the length and breadth of each table and find the ratio amongst them. 

(b) Find for which table the ratio of the length and breadth is the greatest.

Solution:

A hypothetical example based on common dimensions of tables. 

Let's say we have three tables:

Table 1: 

Length = 120 cm 

Breadth = 80 cm

Table 2: 

Length = 150 cm 

Breadth = 90 cm

Table 3: 

Length = 100 cm 

Breadth = 70 cm

(a) Ratio of length to breadth for each table: 

Table 1:  120 80  = 1.5

Table 2:  150 90  = 1.67

Table 3:  100 70  = 1.43

(b) Comparing the ratios: The greatest ratio of length to breadth is for Table 2, with a ratio of 1.67. Therefore, Table 2 has the greatest ratio of length to breadth.

E.5.5. Do you know any story or incident where the word ‘ratio’ has been used? Or have you seen the word ‘ratio’ or the symbol ‘:’ written anywhere? Find some real incidents and draw pictures or describe how and where you found them and tell your teachers and classmates.

Solution:

Examples of where you might encounter these in real life:

Grocery Store: In a grocery store, you might see a sign indicating a price ratio such as "3:1" for a sale like "Buy 3, Get 1 Free". This means for every three items you purchase, you get one free.

Recipes: Many recipes use ratios to indicate ingredient proportions. For instance, a recipe for pancakes might call for a ratio of 2 cups of flour to 1 cup of milk, which is written as "2:1". 

You can imagine encountering these instances in your daily life, whether it's in a store, kitchen, financial report, architectural plan, or educational materials.

E.5.6. Search and find some examples around your real life or you heard about, where same types or similar two quantities have been compared, but their units were different. Then describe how the units were converted to the same unit.

Solution:

Answer will be posted soon. Please wait...

[Solve the following problems: Page 137]

E.5.1. Simplify the following ratios: 

(a) 9 : 12 

(b) 15 : 21 

(c) 45 : 36 

(d) 65 : 26

Solution:

Simplify ratios, we divide both parts of the ratio by their greatest common divisor (GCD). 

Here are the simplified ratios:

(a) 9 : 12 

To simplify, find the greatest common divisor of 9 and 12, which is 3. Then, divide both parts by

93 : 12 3 = 3 : 4

(b) 15 : 21 

The greatest common divisor of 15 and 21 is 3. Divide both parts by 3: 

15 3 : 21 3 = 5 : 7

(c) 45 : 36 

The greatest common divisor of 45 and 36 is 9. Divide both parts by 9: 

45 9 : 36 9 = 5 : 4

(d) 65 : 26 

The greatest common divisor of 65 and 26 is 13. Divide both parts by 13: 

6513 : 2613 = 5 : 2 

So, the simplified ratios are: 

(a) 3 : 4 

(b) 5 : 7 

(c) 5 : 4 

(d) 5 : 2

E.5.2. Identify the equivalent ratios below: 

12 : 18;   

6 : 18;   

15 : 10;   

3 : 2;   

1 : 3;   

2 : 6;   

12 : 8

Solution:

To identify equivalent ratios, we need to find ratios that represent the same relationship or proportionality. This can be done by simplifying each ratio to its simplest form and then comparing them. Let's simplify each ratio: 

12:18 can be simplified by dividing both parts by their greatest common divisor, which is 6: 

12 6 : 18 6 = 2 : 3

6 : 18 can be simplified by dividing both parts by their greatest common divisor, which is 6: 

6 6 : 18 6 = 1 : 3

15:10 can be simplified by dividing both parts by their greatest common divisor, which is 5: 

15 5 : 10 5 = 3 : 2

3:2 is already in its simplest form.1:3 is already in its simplest form. 

2:6 can be simplified by dividing both parts by their greatest common divisor, which is 2: 

22 : 62 = 1 : 3

12:8 can be simplified by dividing both parts by their greatest common divisor, which is 4: 

12 4 : 84 = 3 : 2

After simplifying, we can see that the following ratios are equivalent:

12:18 is equivalent to 2:3 

6:18 is equivalent to 1:3 

15:10 is equivalent to 3:2 

2:6 is equivalent to 1:3 

12:8 is equivalent to 3:2

E.5.3. In a school, there are 450 boys and 500 girls. Write the ratio of the boys and girls of the school in the simplest form.

Solution:

To find the simplest form of the ratio of boys to girls in the school, we need to divide both numbers by their greatest common divisor (GCD). 

The GCD of 450 and 500 is 50. So, we divide both numbers by 50:

450 ÷ 50 = 9 

500 ÷ 50 = 10 

Therefore, the simplest form of the ratio of boys to girls is 9:10

E.5.4. Fill up the empty boxes of the following equivalent ratios: 

(a) 2 : 3 = 8 : 

(b) 5 : 6 =  : 36 

(c) 7 :  = 42 : 54 

(d)  : 9 = 63 : 81

Solution:

Answer will be posted soon. Please wait...

E.5.5. The ratio of the width and length of a hall room is 2 : 5. Fill up the following table with possible values of the width and length:

Solution:

Answer will be posted soon. Please wait...