Points to Remember:
- This is a mathematical word problem requiring the application of ratios and proportions.
- We need to find the individual amounts received by X, Y, and Z based on the given ratios.
Introduction:
This question involves a simple ratio problem. We are given that a total sum of â¹510 is divided among three individuals, X, Y, and Z, with specific ratios defining their shares. Solving this requires translating the verbal description into algebraic equations and then solving for the unknown variables representing the amounts each person receives.
Body:
1. Defining the Ratios:
The problem states:
- X gets (2/3) of what Y gets. This can be written as: X = (2/3)Y
- Y gets (1/4) of what Z gets. This can be written as: Y = (1/4)Z
2. Expressing all variables in terms of Z:
To solve this, it’s easiest to express X and Y in terms of Z. We already have Y = (1/4)Z. Substituting this into the equation for X, we get:
X = (2/3) * (1/4)Z = (2/12)Z = (1/6)Z
Now we have all three variables (X, Y, Z) expressed in terms of Z.
3. Solving for Z:
The total amount is â¹510. Therefore:
X + Y + Z = 510
Substituting the expressions from step 2:
(1/6)Z + (1/4)Z + Z = 510
To solve this, we find a common denominator (12):
(2/12)Z + (3/12)Z + (12/12)Z = 510
(17/12)Z = 510
Z = (510 * 12) / 17 = 30 * 12 = 360
Therefore, Z receives â¹360.
4. Solving for X and Y:
Now that we know Z, we can easily find X and Y:
Y = (1/4)Z = (1/4) * 360 = 90
X = (1/6)Z = (1/6) * 360 = 60
5. Summary of Results:
- X receives â¹60
- Y receives â¹90
- Z receives â¹360
Conclusion:
In conclusion, by translating the given ratios into algebraic equations and solving them systematically, we determined that X receives â¹60, Y receives â¹90, and Z receives â¹360. This demonstrates the importance of carefully defining variables and using appropriate algebraic techniques to solve word problems involving ratios and proportions. The solution highlights the power of expressing all variables in terms of a single variable to simplify the solving process. This approach ensures accuracy and efficiency in solving similar problems involving the distribution of resources or amounts based on defined ratios.
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