Solving the Fraction Problem
We need to find a fraction based on two conditions.
Setting Up the Equations
Let the fraction be represented as $\frac{x}{y}$, where $x$ is the numerator and $y$ is the denominator.
- Condition 1: The sum of the numerator and denominator is 11. $x + y = 11$
- Condition 2: If the numerator is decreased by 1, the fraction becomes $\frac{1}{4}$. $\frac{x-1}{y} = \frac{1}{4}$
Solving for the Numerator and Denominator
We can solve this system of equations. First, express $y$ from the first equation:
$y = 11 - x$Now, substitute this expression for $y$ into the second equation:
$\frac{x-1}{11-x} = \frac{1}{4}$Cross-multiply to solve for $x$:
$4(x-1) = 1(11-x)$ $4x - 4 = 11 - x$Combine like terms:
$4x + x = 11 + 4$ $5x = 15$ $x = \frac{15}{5}$ $x = 3$Now, find the value of $y$ using $y = 11 - x$:
$y = 11 - 3$ $y = 8$Finding the Fraction
The numerator ($x$) is 3 and the denominator ($y$) is 8. Therefore, the fraction is:
$\frac{3}{8}$Verification
- Sum of numerator and denominator: $3 + 8 = 11$.
- If numerator is decreased by 1: $\frac{3-1}{8} = \frac{2}{8}$, which simplifies to $\frac{1}{4}$.
Both conditions are satisfied.
Similar Questions
- The numerator of a fraction is increased by $20\%$ and the denominator is decreased by $50\%$. If the resultant fraction is $\frac{5}{6}$, then what will be the original fraction?
- $1.236576576...$
can be written in the form of: - What smallest fraction should be added to $3\frac{2}{3} + 6\frac{7}{12} + 4\frac{9}{36} + 5 + 7\frac{1}{12}$ to make the sum a whole number?
- What is the positive difference between $\frac{5}{16}$ and its reciprocal?
- How much does one need to add to $\frac{1}{2}$ to get $2$?
- Solve the following:
$(0.\overline{5} + 0.\overline{6} + 0.\overline{7} + 0.\overline{8}) = ?$ - What is the sum of $\frac{1}{3}$, $\frac{4}{3}$ and $\frac{3}{4}$?
- Four persons A, B, C and D completed a task in $\frac{2}{3} \text{ h}, \frac{3}{4} \text{ h}, \frac{4}{5} \text{ h}$ and $\frac{1}{5} \text{ h}$ respectively. Who among the following took the highest amount of time to complete the task?
- If the numerator of a fraction is increased by 30% and its denominator is decreased by 35%, the value of the fraction becomes $\frac{3}{15}$. Find the original fraction.
- Two-fifth of seven-seventeenth of three-fifth of a number is 84. Find 40% of that number.
Important Questions from Fractions
Which fraction among the following is the least ?
\(\frac{5}{11}, \frac{7}{12}, \frac{8}{13}, \frac{9}{17}\)
Find the value of the following expression:
\(\frac{{3 \div 1 \times 2 + 5 - 2}}{{3 \times 3 - 2}}\)
Simplify the expression 441 ÷ \(\left[270 \div \frac{3}{7}+\left(17\div \frac{1}{3}\right)-\left(8\frac{1}{2}-\frac{5}{2}\right)\right]\)
If the sum of two positive numbers is 65 and the square root of their product is 26, then the sum of their reciprocals is:
The value of \(9 \div [\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}\div(\frac{3}{4}-\frac{1}{3})\;of\;\frac{2}{9}]\) is:
