Solving the Proportion Problem: Finding the Value of x

The question provides a proportion relating four quantities: 0.75, x, 2.5, and 8. A proportion states that two ratios are equal. The given proportion is written as 0.75 : x :: 2.5 : 8. The symbol '::' represents equality between the two ratios, meaning 0.75 : x is equal to 2.5 : 8.

We can write a proportion as a fraction equation:

If a : b :: c : d, then $\frac{a}{b} = \frac{c}{d}$.

Applying this to our problem, 0.75 : x :: 2.5 : 8 becomes:

$\frac{0.75}{x} = \frac{2.5}{8}$

Steps to Solve for x in the Proportion

To find the value of x, we can use the method of cross-multiplication. Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the products equal.

For $\frac{0.75}{x} = \frac{2.5}{8}$, cross-multiplication gives us:

$0.75 \times 8 = x \times 2.5$

Now, we need to perform the multiplication on both sides of the equation:

• Calculate the left side: $0.75 \times 8$.

  We can think of 0.75 as $\frac{3}{4}$. So, $0.75 \times 8 = \frac{3}{4} \times 8 = 3 \times \frac{8}{4} = 3 \times 2 = 6$.

  Alternatively, multiplying 75 by 8 gives 600. Since 0.75 has two decimal places, the result will also have two decimal places, giving 6.00 or simply 6.

• The equation becomes: $6 = 2.5x$.

The next step is to isolate x. To do this, we divide both sides of the equation by 2.5 (the coefficient of x):

$x = \frac{6}{2.5}$

To divide by a decimal, it's often easier to remove the decimal by multiplying both the numerator and the denominator by a power of 10. In this case, multiplying by 10 will make 2.5 into 25:

$x = \frac{6 \times 10}{2.5 \times 10} = \frac{60}{25}$

Now, simplify the fraction $\frac{60}{25}$: both 60 and 25 are divisible by 5.

$x = \frac{60 \div 5}{25 \div 5} = \frac{12}{5}$

Finally, convert the fraction $\frac{12}{5}$ to a decimal:

$x = 12 \div 5 = 2.4$

So, the value of x that satisfies the proportion 0.75 : x :: 2.5 : 8 is 2.4.

Let's verify the answer by substituting x = 2.4 back into the proportion:

0.75 : 2.4 :: 2.5 : 8

This means $\frac{0.75}{2.4}$ should be equal to $\frac{2.5}{8}$.

• Ratio 1: $\frac{0.75}{2.4} = \frac{0.75 \times 100}{2.4 \times 100} = \frac{75}{240}$. Both are divisible by 15: $\frac{75 \div 15}{240 \div 15} = \frac{5}{16}$.
• Ratio 2: $\frac{2.5}{8} = \frac{2.5 \times 10}{8 \times 10} = \frac{25}{80}$. Both are divisible by 5: $\frac{25 \div 5}{80 \div 5} = \frac{5}{16}$.

Since both ratios are equal to $\frac{5}{16}$, the value x = 2.4 is correct.

Revision Table: Key Concepts

Additional Information on Ratios and Proportions

Ratios and proportions are fundamental concepts in mathematics used to compare quantities and solve problems involving scaling and relationships between numbers. Understanding how to set up and solve proportions is crucial for various topics, including percentages, similar figures, scale drawings, and rates.

• Terms of a Proportion: In the proportion a : b :: c : d, a and d are called the 'extremes', and b and c are called the 'means'. The property of proportions states that the product of the extremes is equal to the product of the means (ad = bc), which is the basis for cross-multiplication.
• Solving for Unknowns: Proportions are often used to find an unknown value when three other values are known. This is exactly what we did in this problem to find x.
• Real-World Applications: Proportions are used widely in cooking (scaling recipes), mapping (scale), construction (ratios of materials), and science (concentration of solutions).