Solving Proportion Problems with Subtraction

This problem involves finding a specific number, let's call it $x$, which when subtracted from a set of four numbers (54, 49, 22, and 21) makes the resulting numbers form a proportion. When four numbers, say $a, b, c, d$, are in proportion, it means the ratio of the first two is equal to the ratio of the last two. Mathematically, this is written as $\frac{a}{b} = \frac{c}{d}$.

Setting up the Proportion Equation

According to the problem, after subtracting $x$ from each number, the new numbers are $(54-x)$, $(49-x)$, $(22-x)$, and $(21-x)$. Since these numbers are in proportion, we can write the equation:

$$ \frac{54 - x}{49 - x} = \frac{22 - x}{21 - x} $$

Solving for the Value of x

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

$$ (54 - x)(21 - x) = (49 - x)(22 - x) $$

Now, we expand both sides of the equation:

$$ (54 \times 21) - (54 \times x) - (x \times 21) + (x \times x) = (49 \times 22) - (49 \times x) - (x \times 22) + (x \times x) $$

Calculate the products:

$$ 1134 - 54x - 21x + x^2 = 1078 - 49x - 22x + x^2 $$

Combine the $x$ terms on each side:

$$ 1134 - 75x + x^2 = 1078 - 71x + x^2 $$

Notice that both sides have an $x^2$ term. We can subtract $x^2$ from both sides:

$$ 1134 - 75x = 1078 - 71x $$

Now, rearrange the terms to isolate $x$. Move the $x$ terms to one side and the constant terms to the other:

$$ 1134 - 1078 = 75x - 71x $$

Perform the subtractions:

$$ 56 = 4x $$

Finally, solve for $x$ by dividing both sides by 4:

$$ x = \frac{56}{4} $$

$$ x = 14 $$

So, the value of $x$ is 14.

Calculating the Required Ratio

The question asks for the ratio of $(8x - 25)$ to $(7x - 26)$. Now that we know $x = 14$, we can substitute this value into the expressions:

First expression: $8x - 25$

$$ 8(14) - 25 = 112 - 25 = 87 $$

Second expression: $7x - 26$

$$ 7(14) - 26 = 98 - 26 = 72 $$

The ratio is the first expression divided by the second expression:

$$ \frac{8x - 25}{7x - 26} = \frac{87}{72} $$

Simplifying the Ratio

We need to simplify the ratio $\frac{87}{72}$ to its simplest form. We find the greatest common divisor (GCD) of 87 and 72.

Factors of 87: 1, 3, 29, 87

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

The greatest common divisor of 87 and 72 is 3.

Divide both the numerator and the denominator by 3:

$$ \frac{87 \div 3}{72 \div 3} = \frac{29}{24} $$

So, the ratio of $(8x - 25)$ to $(7x - 26)$ is 29 : 24.

Summary of Steps

• Set up the proportion equation based on the problem statement.
• Solve the equation for the variable $x$ using cross-multiplication and algebraic manipulation.
• Substitute the found value of $x$ into the expressions for the ratio.
• Calculate the values of the expressions.
• Form the ratio and simplify it to its lowest terms.

Revision Table: Key Concepts in Proportion and Ratio

Additional Information on Proportion Problems

Proportion problems often appear in various forms in mathematics. Understanding the fundamental principle that the ratio between the first pair of numbers is equal to the ratio between the second pair is key. When a variable is involved, especially when it's being added to or subtracted from the numbers, the problem becomes an algebraic equation that needs to be solved.

Common variations of this type of problem include:

• Finding a number to add to a set of numbers to make them proportional.
• Finding a number to multiply or divide each number by to make them proportional.
• Problems involving mean proportionals (where the middle two terms are the same, e.g., $a:b::b:c$).
• Word problems applying proportion concepts to real-life scenarios like scaling, mixing ratios, or rates.

Solving these problems relies on setting up the correct algebraic expression based on the definition of proportion and then using standard algebraic techniques to solve for the unknown variable. Always remember to check if the final ratio needs to be simplified.