Understanding the Fourth Proportional

The question asks us to find the fourth proportional to the numbers 1.5, 4.8, and 8. Let's first understand what the term 'fourth proportional' means.

In a proportion, if four quantities a, b, c, and d are such that the ratio of the first two (a:b) is equal to the ratio of the last two (c:d), then they are said to be in proportion. This is written as a : b :: c : d, or mathematically, $\frac{a}{b} = \frac{c}{d}$. In this relationship, 'd' is called the fourth proportional to a, b, and c.

Setting up the Proportion

Given the numbers 1.5, 4.8, and 8, we need to find a fourth number, let's call it 'x', such that these four numbers are in proportion in that order. This means:

$1.5 : 4.8 :: 8 : x$

This can be written as a fraction equation:

$\frac{1.5}{4.8} = \frac{8}{x}$

Solving for the Fourth Proportional (x)

To find the value of 'x', we can use cross-multiplication. The product of the means (inner terms) is equal to the product of the extremes (outer terms).

Product of extremes = $1.5 \times x$

Product of means = $4.8 \times 8$

So, we have the equation:

$1.5x = 4.8 \times 8$

First, calculate the product on the right side:

$4.8 \times 8 = 38.4$

Now the equation is:

$1.5x = 38.4$

To find 'x', divide both sides by 1.5:

$x = \frac{38.4}{1.5}$

To make the division easier, we can remove the decimal points by multiplying the numerator and the denominator by 10:

$x = \frac{38.4 \times 10}{1.5 \times 10} = \frac{384}{15}$

Now, perform the division:

$384 \div 15$

We can simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 3:

$\frac{384 \div 3}{15 \div 3} = \frac{128}{5}$

Now, convert the fraction to a decimal:

$\frac{128}{5} = 25.6$

So, the fourth proportional to 1.5, 4.8, and 8 is 25.6.

Verification

Let's check if the ratio holds with x = 25.6:

$\frac{1.5}{4.8}$ and $\frac{8}{25.6}$

$\frac{1.5}{4.8} = \frac{15}{48} = \frac{5}{16}$ (dividing numerator and denominator by 3)

$\frac{8}{25.6} = \frac{80}{256}$ (multiplying numerator and denominator by 10)

Now, simplify $\frac{80}{256}$. Divide both by 16:

$\frac{80 \div 16}{256 \div 16} = \frac{5}{16}$

Since $\frac{1.5}{4.8} = \frac{5}{16}$ and $\frac{8}{25.6} = \frac{5}{16}$, the ratios are equal, confirming that 25.6 is the correct fourth proportional.

Final Answer

The fourth proportional to 1.5, 4.8 and 8 is 25.6.

Revision Table: Key Concepts

Additional Information: Ratio and Proportion Properties

• If a, b, c, d are in proportion, then $ad = bc$. This is known as the product of extremes equals the product of means.
• This property is used extensively to solve problems involving proportion, like finding missing terms (mean proportional, third proportional, or fourth proportional).
• A mean proportional to 'a' and 'c' is 'b' such that a : b :: b : c, meaning $b^2 = ac$ or $b = \sqrt{ac}$.
• A third proportional to 'a' and 'b' is 'c' such that a : b :: b : c, meaning $ac = b^2$ or $c = \frac{b^2}{a}$.