Test Score Problem Analysis
This problem involves finding the ratio between the scores of two students, X and Y, based on given conditions relating their scores. We need to use algebra to solve for the unknown scores and their ratio.
- Let \(S_X\) denote the score of student X.
- Let \(S_Y\) denote the score of student Y.
Score Conditions Breakdown
The problem provides two key pieces of information that we can translate into mathematical equations:
- "The score of X is 20 more than that of Y." This translates to the equation:
\(S_X = S_Y + 20\)
- "The score of X is 75% of the sum of the scores of X and Y." This translates to:
\(S_X = 75\% \times (S_X + S_Y)\)
To make calculations easier, we convert the percentage to a fraction:
\(S_X = \frac{75}{100} \times (S_X + S_Y)\)
Simplifying the fraction gives:
\(S_X = \frac{3}{4} (S_X + S_Y)\)
Calculating the Ratio of Scores
The second equation gives us a direct relationship between \(S_X\) and \(S_Y\). Let's use it to find this relationship.
- Start with the equation derived from the percentage condition:
\(S_X = \frac{3}{4} (S_X + S_Y)\)
- To eliminate the fraction, multiply both sides of the equation by 4:
\(4 \times S_X = 3 \times (S_X + S_Y)\)
Distribute the 3 on the right side:
\(4S_X = 3S_X + 3S_Y\)
- Now, rearrange the terms to group \(S_X\) terms together. Subtract \(3S_X\) from both sides:
\(4S_X - 3S_X = 3S_Y\)
This simplifies to:
\(S_X = 3S_Y\)
This equation, \(S_X = 3S_Y\), directly shows that the score of X is three times the score of Y. This means the ratio \(S_X : S_Y\) is \(3:1\).
Although we have found the ratio, we can use the first condition (\(S_X = S_Y + 20\)) to find the specific scores and verify our result.
- Substitute the relationship \(S_X = 3S_Y\) into the first equation:
\(3S_Y = S_Y + 20\)
- Solve for \(S_Y\). Subtract \(S_Y\) from both sides:
\(3S_Y - S_Y = 20\)
\(2S_Y = 20\)
- Divide by 2:
\(S_Y = \frac{20}{2}\)
\(S_Y = 10\)
- Now, find \(S_X\) using the relationship \(S_X = 3S_Y\):
\(S_X = 3 \times 10\)
\(S_X = 30\)
So, student X scored 30 and student Y scored 10. Let's quickly check if these scores fit the original problem statements:
- Is X's score (30) 20 more than Y's score (10)? Yes, \(30 = 10 + 20\).
- Is X's score (30) 75% of the sum of scores (30 + 10 = 40)? Yes, \(0.75 \times 40 = 30\).
Both conditions are met.
Final Score Ratio Determination
The question asks for the ratio of the score of X to the score of Y, which is \(S_X : S_Y\).
Using the calculated scores, the ratio is \(30 : 10\).
To simplify the ratio, we divide both parts by their greatest common divisor, which is 10:
\(\frac{30}{10} : \frac{10}{10}\)
\(3 : 1\)
Alternatively, the relationship \(S_X = 3S_Y\) directly gives us the ratio \(3:1\).
Similar Questions
- If \(P^2\) varies as R and \(Q^2\) varies as R, \((P \ne Q)\), then which of the following are correct?
1. \(P^2 + Q^2\) varies as R.
2. PQ varies as R.
3. \(P^2 - Q^2\) varies as R.
Select the correct answer using the code given below: - If \(\frac{p+q}{q+r}=\frac{r+s}{s+p}\); \((q+r) \neq 0\), \((s+p) \neq 0\), then which one of the following is correct?
If \(a:b:c:d = \sqrt{4}: \sqrt{3}: \sqrt{2}: \sqrt{1}\), then what is the value of \(\frac{(-a^2 + b^2 + c^2 + d^2)}{(a^2 - b^2 + c^2 - d^2)}\) ?
- If \(m: n = 1:2\) and \(p:q=3:4\), then what is \((2m + 4p): (n + 3q)\) equal to ?
- If \((a+b): (b+c): (c+a) = 5:7:6\), then what is the value of \((a-b+c):(a+b-c)\)?
- ₹ 9400 is distributed among P, Q, R in such a way that if ₹ 93, ₹ 24, ₹ 55 are deducted from their respective shares, then they have money in the ratio \(3:4:5\). What is the share of P?
\(p\) varies directly as \((x^2 + y^2 + z^2)\). When \(x = 1, y = 2, z = 3\), then \(p = 70\). What is the value of \(p\) when \(x = -1, y = 1\), \(z = 5\)?
- A locomotive engine can go 40 km/hr. Its speed gets reduced by a quantity that varies directly as the square root of the number of wagons attached. It is known that its speed becomes 16 km/hr if 36 wagons are attached. What is the smallest number of wagons with which the engine is unable to move ?
- If \(x\) varies directly as \(y\), then which of the following is/are correct ?
I. \((x^2+y^2)\) varies directly as \(y^2\).
II. \(\left(\frac{x^3}{y^2}\right)\) varies directly as \(y\).
Select the answer using the code given below : - Two numbers P and Q are such that P : Q = x : y. If 1 is added to both the numbers, the ratio becomes 7 : 10. If 3 is added to both the numbers, the ratio becomes 3 : 4. What is (x + y) : (x - y) equal to ?
Important Questions from Ratio and proportion
A sum of Rs. 2,500 is distributed among X, Y and Z in the ratio 1/2 : 3/4 : 5/6 . What is the difference between the maximum share and the minimum share?
In a class, there are three groups A, B and C. If one student from group A and two students from group B are shifted to group C, then what happens to the average weight of the students of the class?
How many different sums can be formed with the denominations Rs. 50, Rs. 100, Rs. 200, Rs. 500 and Rs. 2,000 taking at least three denominations at a time?
In a 500 metres race, B starts 45 metres ahead of A, but A wins the race while B is still 35 metres behind. What is the ratio of the speeds of A to B assuming that both start at the same time?
The monthly incomes of Peter and Paul are in the ratio of 4 : 3. Their expenses are in the ratio of 3 : 2. If each saves Rs. 6,000 at the end of the month, their monthly incomes respectively are (in Rs.)
