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
Six years ago, the ratio of ages of A to B was 7 ∶ 5. After 4 years from now, the ratio of their ages will be 11 ∶ 9. What is A’s age at present?
7p - [3q - {8p - (4q - 10p)}] = ?
Surya is 25 years older than his son. In 5 years, he will be twice as old as his son. What will be Surya’s age after 3 years?
The difference between the ages of two sisters is 2 years when father’s age is 52. Father is elder by 2 years to mother. Elder sister’s age is half of mother’s age. Find the age of younger sister?
The sum of the digits of a 2 digit number is 9, When 27 is added to the number, the digits get interchanged. Find the number.
A. 45
B. 36
C. 18
D. 27
