Solving Ratio Problems: Algebraic Expressions
This problem requires us to simplify an algebraic expression involving ratios. We are given two initial ratios and asked to find the ratio of a combined expression.
Understanding the Given Ratios
We are provided with the following ratios:
- \(m: n = 1:2\)
- \(p: q = 3:4\)
From these ratios, we can express the relationships between the variables algebraically:
- From \(m: n = 1:2\), we can write \(\frac{m}{n} = \frac{1}{2}\). This implies \(n = 2m\).
- From \(p: q = 3:4\), we can write \(\frac{p}{q} = \frac{3}{4}\). This implies \(q = \frac{4}{3}p\).
Calculating the Target Expression
The expression we need to find the ratio for is \((2m + 4p): (n + 3q)\).
Let's substitute the expressions for \(n\) and \(q\) that we found into the second part of the ratio \((n + 3q)\):
- Substitute \(n = 2m\): The first term becomes \(2m\).
- Substitute \(q = \frac{4}{3}p\): The second term becomes \(3q = 3 \times (\frac{4}{3}p)\).
Now, let's simplify the second part of the expression:
\(n + 3q = (2m) + 3 \times (\frac{4}{3}p)\)
Simplify the multiplication:
\(n + 3q = 2m + \frac{12}{3}p\)
\(n + 3q = 2m + 4p\)
Determining the Final Ratio
Now we have the two parts of the ratio we need to find:
- The first part is \((2m + 4p)\).
- The second part simplifies to \((n + 3q) = (2m + 4p)\).
Therefore, the ratio \((2m + 4p): (n + 3q)\) becomes:
\((2m + 4p) : (2m + 4p)\)
Any quantity compared to itself in a ratio is \(1:1\). Assuming \(2m + 4p\) is not equal to zero (which is usually the case with positive ratios), the ratio simplifies to \(1:1\).
Final Answer: The final answer is 1:1
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 \((a+b): (b+c): (c+a) = 5:7:6\), 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 : - 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.)
