A sum of Rs. 120 is divided among p, q and r in such a way that p gets Rs. 9 more than q and q gets Rs. 6 more than r. what is the ratio of shares of p, q and r respectively?
16 ∶ 13 ∶ 11
Understanding the Problem: Dividing a Sum and Finding Ratio
The question asks us to determine the ratio of the shares of three individuals, p, q, and r, when a total sum of Rs. 120 is divided among them based on specific conditions regarding how their shares relate to each other.
We are given two key pieces of information about the distribution:
- p gets Rs. 9 more than q.
- q gets Rs. 6 more than r.
We need to find the ratio p : q : r.
Setting Up the Shares with Variables
To solve this, we can express the shares of p and q in terms of r's share, as the conditions link them sequentially:
- Let r's share be represented by \(x\) rupees.
- Since q gets Rs. 6 more than r, q's share is \(x + 6\) rupees.
- Since p gets Rs. 9 more than q, p's share is \((q\text{'s share}) + 9\). Substituting q's share, p's share is \((x + 6) + 9 = x + 15\) rupees.
Forming and Solving the Equation
The total sum distributed among p, q, and r is Rs. 120. This means the sum of their individual shares must equal the total sum:
Share of p + Share of q + Share of r = Total Sum
Substituting the expressions for their shares:
\[(x + 15) + (x + 6) + x = 120\]Now, let's solve this equation for \(x\):
\[x + 15 + x + 6 + x = 120\] \[3x + 21 = 120\]Subtract 21 from both sides:
\[3x = 120 - 21\] \[3x = 99\]Divide both sides by 3:
\[x = \frac{99}{3}\] \[x = 33\]Calculating Individual Shares
Now that we have the value of \(x\), which represents r's share, we can find the shares of p and q:
- r's share = \(x = 33\) rupees.
- q's share = \(x + 6 = 33 + 6 = 39\) rupees.
- p's share = \(x + 15 = 33 + 15 = 48\) rupees.
Let's quickly check if the sum of these shares is indeed Rs. 120:
\[48 + 39 + 33 = 87 + 33 = 120\]The shares add up correctly to the total sum.
Determining the Ratio of Shares
The question asks for the ratio of shares of p, q, and r respectively, which is p : q : r.
Using the calculated shares, the ratio is:
\[48 : 39 : 33\]To simplify this ratio, we need to find the greatest common divisor (GCD) of 48, 39, and 33. All three numbers are divisible by 3.
- \(48 \div 3 = 16\)
- \(39 \div 3 = 13\)
- \(33 \div 3 = 11\)
So, the simplified ratio p : q : r is 16 : 13 : 11.
| Individual | Share (in terms of x) | Calculated Share (in Rs.) |
|---|---|---|
| r | \(x\) | 33 |
| q | \(x + 6\) | 39 |
| p | \(x + 15\) | 48 |
The ratio of shares of p, q, and r is 16 : 13 : 11.
Revision Table: Key Concepts for Ratio Problems
| Concept | Description | Application in this Problem |
|---|---|---|
| Representing Unknowns | Using variables (like \(x\)) to represent unknown quantities. | Used \(x\) for r's share, then expressed q's and p's shares in terms of \(x\). |
| Forming Equation | Setting up an algebraic equation based on the problem's conditions. | Sum of individual shares equals the total sum: \((x+15) + (x+6) + x = 120\). |
| Solving Linear Equation | Finding the value of the variable that satisfies the equation. | Solved \(3x + 21 = 120\) to find \(x = 33\). |
| Calculating Values | Substituting the variable's value back into the expressions to find specific quantities. | Calculated the shares of p, q, and r using \(x=33\). |
| Simplifying Ratio | Dividing all terms in a ratio by their GCD to express it in its simplest form. | Simplified \(48:39:33\) by dividing by 3 to get \(16:13:11\). |
Additional Information: Ratio and Proportion Basics
A ratio is a comparison of two or more quantities of the same kind. It is usually written using a colon (:) or as a fraction.
For example, the ratio 16 : 13 : 11 means that for every 16 units p receives, q receives 13 units, and r receives 11 units. The actual value of a unit depends on the total sum being divided.
In this problem, the total parts in the ratio are \(16 + 13 + 11 = 40\). The total sum is Rs. 120. This means each 'part' of the ratio corresponds to \(\frac{120}{40} = 3\) rupees.
Using this information, we can also calculate the shares directly from the ratio:
- p's share = 16 parts = \(16 \times 3 = 48\) rupees.
- q's share = 13 parts = \(13 \times 3 = 39\) rupees.
- r's share = 11 parts = \(11 \times 3 = 33\) rupees.
This confirms our previous calculation and is another way to think about ratio distribution problems.
Important Questions from Simple Ratios
If the value of \(\frac{{p - q}}{q} = \frac{4}{3}\) .find p : q
Half of the villagers of a certain village have their own houses. One-fifth of the villagers cultivate paddy. One-third of the villagers are literate. Four-fifth of the villagers are under 25 years of age. Which one of the following statements is certainly correct ?
The two numbers are in the ratio of 9 ∶ 7 and the difference between of these two number is 6000. What is the sum of the two numbers?
A certain number is added to each of a pair of numbers which are in the ratio 4 ∶ 5. The sum of the resulting numbers is 39 and their ratio (taken in the same order as mentioned above) is 6 ∶ 7. What is the number added?
If (p + q) ∶ (q + r) ∶ (r + p) = 5 ∶ 6 ∶ 7 and p + q + r = 18, find the value of p ∶ q ∶ r
