Find the third proportional to $6\frac{1}{4}$ and 5

Understand the Problem

The question is asking to find the third proportional to the numbers $6\frac{1}{4}$ and 5. In a proportion $a:b::c:d$, where $a, b, c, d$ are in proportion, the third proportional to $a$ and $b$ is the value $x$ such that $a:b::b:x$. In this case, we need to find $x$ such that $6\frac{1}{4} : 5 :: 5 : x$.

Answer

(c) 4

Steps to Solve

Set up the proportion

We are given $a = 6\frac{1}{4}$ and $b = 5$. We need to find $x$ such that $a:b::b:x$, which means $\frac{a}{b} = \frac{b}{x}$.

Convert the mixed fraction to an improper fraction

$6\frac{1}{4} = \frac{6 \times 4 + 1}{4} = \frac{24 + 1}{4} = \frac{25}{4}$

Substitute the values of $a$ and $b$ into the proportion equation

$\frac{\frac{25}{4}}{5} = \frac{5}{x}$

Simplify the left side of the equation $\frac{25}{4} \div 5 = \frac{25}{4} \times \frac{1}{5} = \frac{25}{20} = \frac{5}{4}$ So, our equation now is: $\frac{5}{4} = \frac{5}{x}$
Solve for $x$

Cross-multiply to get $5x = 5 \times 4$, therefore $5x = 20$. Finally, divide both sides by 5: $x = \frac{20}{5} = 4$.

More Information

The third proportional to $6\frac{1}{4}$ and 5 is 4.

Tips

A common mistake is to incorrectly set up the proportion or to make errors when simplifying the fractions. Remembering the definition of third proportional is crucial to setting up the problem correctly i.e. $a:b::b:x$. Watch out for arithmetic errors when dealing with the fractions.

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