Let the two numbers be $x$ and $y$.
Setting Up the Equations
From the problem statement, we can form two linear equations:
- The sum of $\frac{1}{3}$ of the first number and $\frac{1}{2}$ of the second number is 8: $ \frac{1}{3}x + \frac{1}{2}y = 8 \quad (\text{Equation 1}) $
- The sum of $\frac{1}{5}$ of the first number and $\frac{1}{6}$ of the second number is 4: $ \frac{1}{5}x + \frac{1}{6}y = 4 \quad (\text{Equation 2}) $
Simplifying the Equations
To solve the system, let's clear the fractions. Multiply Equation 1 by the least common multiple of 3 and 2, which is 6:
$ 6 \left( \frac{1}{3}x + \frac{1}{2}y \right) = 6 \times 8 $ $ 2x + 3y = 48 \quad (\text{Equation 1'}) $Multiply Equation 2 by the least common multiple of 5 and 6, which is 30:
$ 30 \left( \frac{1}{5}x + \frac{1}{6}y \right) = 30 \times 4 $ $ 6x + 5y = 120 \quad (\text{Equation 2'}) $Solving for the Numbers
Now we have a system of linear equations:
- $2x + 3y = 48$
- $6x + 5y = 120$
We can use the elimination method. Multiply Equation 1' by 3 to make the coefficients of $x$ match:
$ 3(2x + 3y) = 3 \times 48 $ $ 6x + 9y = 144 \quad (\text{Equation 3}) $Subtract Equation 2' from Equation 3:
$ (6x + 9y) - (6x + 5y) = 144 - 120 $ $ 4y = 24 $ $ y = \frac{24}{4} $ $ y = 6 $Substitute the value of $y = 6$ back into Equation 1':
$ 2x + 3(6) = 48 $ $ 2x + 18 = 48 $ $ 2x = 48 - 18 $ $ 2x = 30 $ $ x = \frac{30}{2} $ $ x = 15 $The two numbers are 15 and 6.
Determining the Largest Number
Compare the two numbers found:
- First number: $x = 15$
- Second number: $y = 6$
The largest of the two numbers is 15.
Similar Questions
- If $2\text{x} + \text{y} = 15$, $\text{y} + 2\text{z} = 17$ and $\text{x} + 2\text{y} = 9$, then the value of $4\text{x} + 3\text{y} + \text{z} = ?$
Ram has ₹1000 in the denomination of ₹10, ₹20 and ₹5 notes. If the number of ₹10 notes is 16 more than that of ₹20 notes and the number of ₹5 notes is twice the number of ₹20 notes, find the total number of notes that Ram has.
An amount of ₹120 is paid using ₹10, ₹5 and ₹2 coins. If the number of coins of ₹10 and ₹2 are interchanged, then the amount becomes ₹304. If the number of ₹5 and ₹2 coins are interchanged, then the amount is ₹165. How many ₹2 coins are used in the payment of ₹120?
Important Questions from Linear Equation in 2 or more Variables
If \(3x+6y+9z = \dfrac{20}{3}, 6x+9y + 3z = \dfrac{17}{3}\) and \(18x+ 27y - z = \dfrac{113}{9}\) , then what is the value of \(75x+113y \ ?\)
If the system of equations 2x - 3y - 3 and -4x + qy - p/2 is inconsistent which of the following cannot be the value of p ?
If \(4x + \dfrac{6}{y}= 15 \) and \(6x - \dfrac{8}{y} = 14\) , then the value of p in y = px - 2 is :
Simplify: 7x + 3x(x – 4) = ?
A. 10x + 12
B. 10x – 12
C. 3x2 + 5x
D. 3x2 – 5xIf 5x + y = 17 and xy = 6, then what is the value of 125x3 + y3 ?
