Example Question - numerator and denominator
Here are examples of questions we've helped users solve.
Solving a Fraction Problem
Let's denote the numerator of the original fraction as \( n \) and the denominator as \( d \). According to the problem, the denominator is four more than the numerator, so: \( d = n + 4 \) The problem also states that if both the numerator and denominator are increased by one, the resulting fraction is simplified to \( \frac{5}{7} \): \( \frac{n + 1}{d + 1} = \frac{5}{7} \) Now we have a system of two equations to solve for the values of \( n \) and \( d \): 1) \( d = n + 4 \) 2) \( \frac{n + 1}{n + 5} = \frac{5}{7} \) (since \( d = n + 4 \), we replaced \( d \) with \( n + 4 \) in the second equation) Next, we can cross-multiply in the second equation to solve for \( n \): \( 7(n + 1) = 5(n + 5) \) Expand both sides: \( 7n + 7 = 5n + 25 \) Subtract \( 5n \) from both sides: \( 2n + 7 = 25 \) Subtract 7 from both sides: \( 2n = 18 \) Divide by 2: \( n = 9 \) Now that we have \( n \), we can find \( d \) using either of the original equations. Let's use the first one: \( d = n + 4 \) Replace \( n \) with 9: \( d = 9 + 4 \) Thus: \( d = 13 \) The original fraction is \( \frac{n}{d} = \frac{9}{13} \).
Simplification of a Complex Fraction
To simplify the given expression, we need to combine the terms in the numerator and the denominator first and then simplify the complex fraction. \[ \frac{\frac{2}{x}-5}{6 + \frac{3}{x}} \] First, let's find a common denominator for the terms in the numerator and the terms in the denominator. For the numerator, the common denominator is \( x \). For the denominator, it is also \( x \). Numerator: \[ \frac{2}{x} - 5 = \frac{2}{x} - \frac{5x}{x} = \frac{2 - 5x}{x} \] Denominator: \[ 6 + \frac{3}{x} = \frac{6x}{x} + \frac{3}{x} = \frac{6x + 3}{x} \] Now that we have a common denominator, we can combine the two parts: \[ \frac{\frac{2 - 5x}{x}}{\frac{6x + 3}{x}} \] Next, we can simplify by multiplying by the reciprocal of the denominator: \[ \frac{2 - 5x}{x} \times \frac{x}{6x + 3} \] Notice that \( x \) in the numerator and denominator will cancel out: \[ \frac{2 - 5x}{6x + 3} \] Now the expression is simplified: \[ \frac{2 - 5x}{6x + 3} \] This is the simplified form of the given expression.
Simplifying Algebraic Expression by Factoring
Certainly! To simplify the given expression, we need to factor out common terms and potentially simplify both the numerator and the denominator. The expression is: (10x^(5n+4) + 125x^(5n+2)) / (3x^(5n+3) - 20x^(5n+1)) Now, let's simplify both parts: The numerator can be factored by x^(5n+2), which is the smallest power of x present in both terms: x^(5n+2)(10x^2 + 125) Similarly, the denominator can be factored by x^(5n+1): x^(5n+1)(3x^2 - 20) After factoring out, we get: x^(5n+2)(10x^2 + 125) / x^(5n+1)(3x^2 - 20) Now, we can reduce the expression by eliminating common factors. Notice that x^(5n+2) in the numerator and x^(5n+1) in the denominator can be simplified: x^(5n+2) / x^(5n+1) = x ((5n+2) - (5n+1)) = x^(1) Now we have: x(10x^2 + 125) / (3x^2 - 20) This cannot be further simplified without knowing the specific values of x or unless the quadratic expressions factor into terms that cancel each other out (which is not the case here). Thus the simplified expression is: x(10x^2 + 125) / (3x^2 - 20)
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