Understanding the Proportion Problem

The question asks us to find a number that, when subtracted from each of the numbers 7, 9, 11, and 15, makes the resulting four numbers proportional. Numbers are in proportion if the ratio of the first two is equal to the ratio of the last two. If four numbers, say $a, b, c, d$, are in proportion, then $\frac{a}{b} = \frac{c}{d}$.

Setting up the Equation

Let the number that is subtracted be $x$. When $x$ is subtracted from 7, 9, 11, and 15, the new numbers are $(7-x)$, $(9-x)$, $(11-x)$, and $(15-x)$.

According to the problem, these resulting numbers are in proportion. Therefore, we can set up the following equation based on the definition of proportion:

$$ \frac{7-x}{9-x} = \frac{11-x}{15-x} $$

Solving for the Subtracted Number

To solve this equation for $x$, we can cross-multiply:

$$ (7-x)(15-x) = (11-x)(9-x) $$

Now, we expand both sides of the equation:

Left side: $(7-x)(15-x) = 7 \times 15 - 7x - 15x + x^2 = 105 - 22x + x^2$

Right side: $(11-x)(9-x) = 11 \times 9 - 11x - 9x + x^2 = 99 - 20x + x^2$

So, the equation becomes:

$$ 105 - 22x + x^2 = 99 - 20x + x^2 $$

Subtract $x^2$ from both sides:

$$ 105 - 22x = 99 - 20x $$

Add $22x$ to both sides:

$$ 105 = 99 - 20x + 22x $$

$$ 105 = 99 + 2x $$

Subtract 99 from both sides:

$$ 105 - 99 = 2x $$

$$ 6 = 2x $$

Divide by 2:

$$ x = \frac{6}{2} $$

$$ x = 3 $$

So, the number subtracted is 3.

Verifying the Subtracted Number

Let's check if subtracting 3 from each number results in a proportion:

• $7 - 3 = 4$
• $9 - 3 = 6$
• $11 - 3 = 8$
• $15 - 3 = 12$

The resulting numbers are 4, 6, 8, and 12.

Let's check if they are in proportion:

Is $\frac{4}{6} = \frac{8}{12}$?

Simplify the fractions:

• $\frac{4}{6}$ simplifies to $\frac{2}{3}$ (dividing numerator and denominator by 2).
• $\frac{8}{12}$ simplifies to $\frac{2}{3}$ (dividing numerator and denominator by 4).

Since $\frac{2}{3} = \frac{2}{3}$, the resulting numbers are indeed in proportion. The number subtracted is 3.

Revision Table: Key Steps for Proportion Problems

Additional Information on Proportion and Ratios

Ratio: A ratio is a comparison of two quantities by division. For example, the ratio of 4 to 6 can be written as $4:6$ or $\frac{4}{6}$. Ratios can often be simplified, just like fractions.

Proportion: A proportion is an equation that states that two ratios are equal. If $\frac{a}{b} = \frac{c}{d}$, then $a, b, c, d$ are in proportion. In a proportion, the product of the means (inner terms) is equal to the product of the extremes (outer terms). In $\frac{a}{b} = \frac{c}{d}$, $b$ and $c$ are the means, and $a$ and $d$ are the extremes. So, $ad = bc$. This property was used when we cross-multiplied to solve the equation.

Problems involving proportion often require setting up an algebraic equation and solving it. It's important to correctly identify the terms that form the ratios based on the problem statement.