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Proportional Reasoning-1 - Sharing, but Not Equally!

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A ratio a:ba:b is used to compare two quantities. In sharing problems, these represent 'parts' of a whole.

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To share a total quantity TT in a given ratio a:ba:b, we first determine the total number of parts by calculating a+ba + b.

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The value of a single part is found by dividing the total quantity by the sum of the ratio terms: Ta+b\frac{T}{a+b}.

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Individual shares are calculated by multiplying the value of one part by the respective ratio term. Share A=a×(Ta+b)A = a \times \left(\frac{T}{a+b}\right) and Share B=b×(Ta+b)B = b \times \left(\frac{T}{a+b}\right).

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The sum of all individual shares must always equal the original total quantity TT.

📐Formulae

Total Parts=a+b+c+…\text{Total Parts} = a + b + c + \dots

Value of One Part=Total QuantityTotal Parts\text{Value of One Part} = \frac{\text{Total Quantity}}{\text{Total Parts}}

Individual Share=Ratio TermSum of Ratio Terms×Total Quantity\text{Individual Share} = \frac{\text{Ratio Term}}{\text{Sum of Ratio Terms}} \times \text{Total Quantity}

💡Examples

Problem 1:

Divide ₹4500₹ 4500 between Rahul and Priya in the ratio 4:54:5.

Solution:

Sum of ratio terms =4+5=9= 4 + 5 = 9. Rahul's share =49×4500=4×500=₹2000= \frac{4}{9} \times 4500 = 4 \times 500 = ₹ 2000. Priya's share =59×4500=5×500=₹2500= \frac{5}{9} \times 4500 = 5 \times 500 = ₹ 2500.

Explanation:

First, we find the total number of parts (99). We then calculate the value of one part (₹500₹ 500) and multiply it by each person's ratio share.

Problem 2:

The angles of a triangle are in the ratio 2:3:52:3:5. Find the measure of the largest angle.

Solution:

We know the sum of angles in a triangle is 180∘180^\circ. Total parts =2+3+5=10= 2 + 3 + 5 = 10. Largest ratio term is 55. Largest angle =510×180∘=12×180∘=90∘= \frac{5}{10} \times 180^\circ = \frac{1}{2} \times 180^\circ = 90^\circ.

Explanation:

The total quantity is the sum of angles (180∘180^\circ). We divide this by the total parts (1010) to find the unit value, then multiply by the largest ratio term (55).

Problem 3:

A total profit of ₹75,000₹ 75,000 is shared between two business partners in the ratio 7:87:8. Find the difference between their shares.

Solution:

Total parts =7+8=15= 7 + 8 = 15. Partner 1 share =715×75000=7×5000=₹35,000= \frac{7}{15} \times 75000 = 7 \times 5000 = ₹ 35,000. Partner 2 share =815×75000=8×5000=₹40,000= \frac{8}{15} \times 75000 = 8 \times 5000 = ₹ 40,000. Difference: 40000−350005000\begin{array}{r} 40000 \\ -35000 \\ \hline 5000 \end{array} Difference is ₹5,000₹ 5,000.

Explanation:

We calculate individual shares based on the total profit and ratio. The difference is found by subtracting the smaller share from the larger one.

Sharing, but Not Equally! Class 8 Notes & Examples