Question

When 5 is added to the numerator and denominator both of a (positive) fraction, then the new ratio of numerator to denominator becomes \(11: 15\) What is the original ratio? (a) \(17: 25\) (b) \(3: 5\) (c) \(28: 40\) (d) none of these

Short answer

Based on the given information, find the original ratio of the positive fraction. The original ratio of the positive fraction is 17 : 25.

Step-by-step solution

Relate the original and new fractions

Let the original fraction be \(\frac{x}{y}\). After adding 5 to both the numerator and denominator, the new fraction becomes \(\frac{x+5}{y+5}\). According to the problem, this new fraction is equal to the ratio \(11 : 15\). Thus, we can write the equation: \(\frac{x+5}{y+5} = \frac{11}{15}\)

Cross-multiply to remove fractions

To eliminate the fractions in our equation, we can cross-multiply: \((x + 5)(15) = (y + 5)(11)\)

Simplify the equation

Now simplify both sides of the equation: \(15x + 75 = 11y + 55\)

Rearrange the equation

Rearrange the equation to have all variables on one side and all constants on the other side: \(15x - 11y = -20\)

Find the original ratio

The given possible solutions are in the form of ratios. To match them to our equation, we can make it homogeneous by dividing both sides by the greatest common divisor of 15 and 11, which is 1. The equation remains the same: \(15x - 11y = -20\) Now check each option by plugging in the given numerator and denominator values for x and y, and see which one satisfies the equation: (a) \(17 : 25\) If \(x = 17\) and \(y = 25\) then: \(15(17) - 11(25) = -20\), which holds true. So the original ratio is \(\boxed{17 : 25}\).

Key concepts

Fractions

Fractions are mathematical expressions representing the division of one quantity by another. They consist of two main parts: the numerator on top and the denominator at the bottom. The fraction \( \frac{x}{y} \) indicates that the number \( x \) is being divided by \( y \). When manipulating or solving problems with fractions, it's crucial to understand how changing the numerator or denominator affects the overall value. Adding a number to both the numerator and the denominator in a fraction, as we see in this problem, alters the ratio between them, hence changing the fraction's value.

• Numerator: The top number in a fraction, indicating how many parts of a whole are being considered.
• Denominator: The bottom number, showing the total number of equal parts the whole is divided into.

Understanding fractions also involves recognizing their equivalencies with ratios. A fraction can be viewed as an expression of a ratio between the two numbers involved. This insight is helpful when attempting to solve problems where the original fraction needs to be modified to meet conditions, such as matching a given ratio like \(11:15\), as seen in the exercise.

Equation Solving

Equation solving involves finding the value of unknown variables that satisfy the condition or equality represented in the equation. In this specific problem, we start with the equation that relates changes in the fraction to a specified ratio: \[ \frac{x+5}{y+5} = \frac{11}{15} \]Through equation solving, our goal is to find the values of \(x\) and \(y\) that will make this equation true. The initial step involves simplifying the equation through cross multiplication, which we will explore in more detail shortly.

• Ensure both sides of the equation maintain balance by following logical and arithmetic rules.
• Rearrange terms to isolate the unknowns, allowing for a clearer path to finding their values.
• Assess each solution provided in multiple choice questions to verify which satisfies the solved equation.

By plugging in various potential values from the problem's choices, you can determine which set of numbers, when substituted back into the equation, keep both sides equal.

Cross Multiplication

Cross multiplication is a mathematical method used to eliminate fractions from an equation by multiplying across the equals sign. This approach is particularly useful when dealing with equations that involve two fractions set equal to each other, like in the problem on hand.Given the equation:\[ \frac{x+5}{y+5} = \frac{11}{15} \]Cross multiplication involves multiplying the numerator of each fraction by the denominator of the other:

• Multiply \( (x+5) \) by \( 15 \)
• Multiply \( (y+5) \) by \( 11 \)

This removes the fractions, resulting in the equation:\[ (x+5) \cdot 15 = (y+5) \cdot 11 \]By expanding these products, we simplify the process of solving the equation as it becomes linear, making it straightforward to isolate and solve for the unknown variables, \(x\) and \(y\). Cross multiplication is a highly effective method when handling proportions and ratios in algebraic equations.