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Solving Ratio Problems with Algebraic Equations
Let the two numbers be P and Q.
The initial ratio is given as P : Q = x : y.
We are given two conditions based on changes to these numbers:
- Condition 1: If 1 is added to both numbers, the ratio becomes 7 : 10. \(\frac{P+1}{Q+1} = \frac{7}{10}\)
- Condition 2: If 3 is added to both numbers, the ratio becomes 3 : 4. \(\frac{P+3}{Q+3} = \frac{3}{4}\)
Deriving the Values of P and Q
Let's convert the conditions into linear equations:
- From Condition 1: \(10(P+1) = 7(Q+1) \\ 10P + 10 = 7Q + 7 \\ 10P - 7Q = -3 \quad \quad (1)\)
- From Condition 2: \(4(P+3) = 3(Q+3) \\ 4P + 12 = 3Q + 9 \\ 4P - 3Q = -3 \quad \quad (2)\)
Now, we solve this system of linear equations. Multiply Equation (2) by 7 and Equation (1) by 3 to eliminate Q:
- \(3 \times (1) \implies 30P - 21Q = -9\)
- \(7 \times (2) \implies 28P - 21Q = -21\)
Subtract the second modified equation from the first:
\((30P - 21Q) - (28P - 21Q) = -9 - (-21) \\ 2P = 12 \\ P = 6\)Substitute P = 6 into Equation (2):
\(4(6) - 3Q = -3 \\ 24 - 3Q = -3 \\ 3Q = 27 \\ Q = 9\)So, the numbers are P = 6 and Q = 9.
Determining the Ratio x : y
The initial ratio P : Q is given as x : y.
Substituting the values P = 6 and Q = 9:
\(x : y = P : Q = 6 : 9\)Simplify the ratio:
\(x : y = 2 : 3\)Therefore, we can take x = 2 and y = 3.
Calculating the Final Expression
The question asks for the value of (x + y) : (x - y).
Substitute x = 2 and y = 3:
\((x + y) : (x - y) = (2 + 3) : (2 - 3) \\ = 5 : (-1) \\ = -5\)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 ?
- Two students X and Y appeared in a test. The score of X is 20 more than that of Y. If the score of X is 75% of the sum of the scores of X and Y, then what is the ratio of score of X to score of Y?
- 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 :
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.)
