Solving Ratio and Proportion Equations

We are given the equation \( \frac{{4a\; + \;7b}}{{4c\; +\; 7d}} = \frac{{4a\; - \;7b}}{{4c\; -\; 7d}} \). Our goal is to find the relationship between \(a, b, c,\) and \(d\) that satisfies this equation.

We can rearrange the terms of the given equation to make it easier to work with. Let's cross-multiply the terms:

\( (4a + 7b)(4c - 7d) = (4a - 7b)(4c + 7d) \)

Now, expand both sides of the equation:

Left side: \( (4a)(4c) + (4a)(-7d) + (7b)(4c) + (7b)(-7d) \)

\( 16ac - 28ad + 28bc - 49bd \)

Right side: \( (4a)(4c) + (4a)(7d) + (-7b)(4c) + (-7b)(7d) \)

\( 16ac + 28ad - 28bc - 49bd \)

So, the equation becomes:

\( 16ac - 28ad + 28bc - 49bd = 16ac + 28ad - 28bc - 49bd \)

Now, let's simplify this equation by moving all terms to one side or canceling common terms. Notice that \(16ac\) and \(-49bd\) appear on both sides with the same sign. We can subtract \(16ac\) and add \(49bd\) to both sides:

\( -28ad + 28bc = 28ad - 28bc \)

Now, let's group the terms involving \(ad\) on one side and terms involving \(bc\) on the other side. Add \(28ad\) to both sides and add \(28bc\) to both sides:

\( 28bc + 28bc = 28ad + 28ad \)

\( 56bc = 56ad \)

Divide both sides by 56:

\( bc = ad \)

We are looking for a ratio relationship between \(a, b, c,\) and \(d\). From \(bc = ad\), we can rearrange the terms to form a ratio. Divide both sides by \(bd\) (assuming \(b \neq 0\) and \(d \neq 0\)) to get the ratio \(\frac{a}{b}\) on one side:

\( \frac{{bc}}{{bd}} = \frac{{ad}}{{bd}} \)

\( \frac{{c}}{{d}} = \frac{{a}}{{b}} \)

This can be written as \( \frac{a}{b} = \frac{c}{d} \).

Alternatively, we could use the property of Componendo and Dividendo. The given equation is \( \frac{{4a\; + \;7b}}{{4c\; +\; 7d}} = \frac{{4a\; - \;7b}}{{4c\; -\; 7d}} \). We can rewrite this by swapping the denominators:

\( \frac{{4a\; + \;7b}}{{4a\; - \;7b}} = \frac{{4c\; +\; 7d}}{{4c\; -\; 7d}} \)

This form looks like \(\frac{X+Y}{X-Y} = \frac{P+Q}{P-Q}\). If this holds, then by the inverse of Componendo and Dividendo, we must have \(\frac{X}{Y} = \frac{P}{Q}\).

In our case, \(X = 4a\), \(Y = 7b\), \(P = 4c\), and \(Q = 7d\). Applying the inverse property:

\( \frac{{4a}}{{7b}} = \frac{{4c}}{{7d}} \)

Now, multiply both sides by \( \frac{7}{4} \) to isolate the ratio \(\frac{a}{b}\):

\( \frac{7}{4} \times \frac{{4a}}{{7b}} = \frac{7}{4} \times \frac{{4c}}{{7d}} \)

\( \frac{{a}}{{b}} = \frac{{c}}{{d}} \)

Both methods yield the same result.

Comparing this result with the given options, we see that the relationship \( \frac{a}{b} = \frac{c}{d} \) matches one of the choices.

Ratio and Proportion Concepts

This problem involves concepts of ratios and proportions. A ratio compares two quantities, and a proportion states that two ratios are equal.

• A ratio is expressed as \(a:b\) or \(\frac{a}{b}\).
• A proportion is an equality between two ratios, such as \(\frac{a}{b} = \frac{c}{d}\).
• In a proportion \( \frac{a}{b} = \frac{c}{d} \), \(a\) and \(d\) are called extremes, and \(b\) and \(c\) are called means. The property \(ad = bc\) (product of extremes equals product of means) is fundamental.

Revision Table: Key Properties Used

Additional Information on Ratio and Proportion

Ratio and proportion are fundamental concepts in mathematics used to compare quantities and express relationships between them. They are widely applied in various fields, including science, engineering, finance, and everyday life (like scaling recipes or maps).

• Continued Proportion: Three numbers \(a, b, c\) are in continued proportion if \( \frac{a}{b} = \frac{b}{c} \). Here, \(b\) is called the mean proportional between \(a\) and \(c\), and \(b^2 = ac\).
• Properties of Proportion: Besides Componendo and Dividendo, other properties include Invertendo (If \( \frac{a}{b} = \frac{c}{d} \), then \( \frac{b}{a} = \frac{d}{c} \)), Alternendo (If \( \frac{a}{b} = \frac{c}{d} \), then \( \frac{a}{c} = \frac{b}{d} \)), and Componendo (If \( \frac{a}{b} = \frac{c}{d} \), then \( \frac{a+b}{b} = \frac{c+d}{d} \)).
• Understanding these properties helps in solving complex problems involving ratios and proportions efficiently, often simplifying algebraic manipulations.