Solving Ratio and Proportion Problems
This problem asks us to find the ratio between two expressions, \((a-b+c)\) and \((a+b-c)\), given the ratios of the sums of pairs of these variables. We are given that:
\( (a+b): (b+c): (c+a) = 5:7:6 \)
Our goal is to determine the value of the ratio \( (a-b+c):(a+b-c) \).
Representing Ratios with a Constant
When dealing with ratios, it's helpful to introduce a constant, let's call it \(k\). We can rewrite the given proportions as equations:
- \( a+b = 5k \)
- \( b+c = 7k \)
- \( c+a = 6k \)
Calculating the Values of Variables
To find the individual values of \(a\), \(b\), and \(c\), we can first add these three equations together:
\( (a+b) + (b+c) + (c+a) = 5k + 7k + 6k \)
Combining like terms, we get:
\( 2a + 2b + 2c = 18k \)
Factor out the 2:
\( 2(a+b+c) = 18k \)
Divide both sides by 2 to find the sum of all three variables:
\( a+b+c = 9k \)
Now, we can find the value of each variable by subtracting the sum of the other two variables from the total sum (\(a+b+c = 9k\)):
- To find \(c\): Subtract the equation \((a+b = 5k)\) from \((a+b+c = 9k)\). \( c = (a+b+c) - (a+b) = 9k - 5k = 4k \)
- To find \(a\): Subtract the equation \((b+c = 7k)\) from \((a+b+c = 9k)\). \( a = (a+b+c) - (b+c) = 9k - 7k = 2k \)
- To find \(b\): Subtract the equation \((c+a = 6k)\) from \((a+b+c = 9k)\). \( b = (a+b+c) - (c+a) = 9k - 6k = 3k \)
So, we have \(a=2k\), \(b=3k\), and \(c=4k\). These are the values of our variables in terms of the constant \(k\). Note that \(k\) cannot be zero, otherwise all sums would be zero, contradicting the given ratios.
Calculating the Required Ratio
Now we need to find the ratio \( (a-b+c):(a+b-c) \). Let's substitute the values we found for \(a\), \(b\), and \(c\):
- Calculate the first part of the ratio, \((a-b+c)\): \( a-b+c = (2k) - (3k) + (4k) = 2k - 3k + 4k = 3k \)
- Calculate the second part of the ratio, \((a+b-c)\): \( a+b-c = (2k) + (3k) - (4k) = 2k + 3k - 4k = k \)
Now, form the ratio:
\( (a-b+c):(a+b-c) = 3k : k \)
Since \(k\) is a non-zero constant, we can simplify this ratio by dividing both parts by \(k\):
\( 3k : k = 3:1 \)
Final Result
The value of the ratio \( (a-b+c):(a+b-c) \) is \( 3:1 \).
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 ?
- ₹ 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 : - 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.)
