7. Which of the following has the largest value?
A $\frac{\left(\frac{1}{2}\right)}{\left(\frac{3}{4}\right)}$
B $\frac{1}{\left(\frac{\left(\frac{2}{3}\right)}{4}\right)}$
C $\frac{\left(\frac{\left(\frac{1}{2}\right)}{3}\right)}{4}$
E $\frac{\left(\frac{1}{\left(\frac{2}{3}\right)}\right)}{4}$
Solution
B
Commentary
The natural method here is to simplify each fraction to the form $\frac{p}{q}$, where $p$ and $q$ are positive integers, and then to look and see which of these simplified fractions has the largest value.

However, if you adopt this method, you will see that just one of the fractions is greater than 1 .

Now, when $x$ and $y$ are positive numbers,
$$
\frac{x}{y}>1 \text { if, and only if } x>y .
$$

It follows that we can answer this question by evaluating the numerator $x$ and the denominator $y$ for each option in turn, and showing that in only one case we have $x>y$. This saves a little work.

In option A the numerator $\frac{1}{2}$ is smaller than the denominator $\frac{3}{4}$ and so the value of the fraction is less than 1 .

In option B the numerator is 1 and the denominator is
$$
\frac{\left(\frac{2}{3}\right)}{4}=\frac{2}{3} \times \frac{1}{4}=\frac{1}{6} .
$$

Since the numerator is greater than the denominator, the value of this fraction is greater than 1 .
In option $\mathrm{C}$ the numerator is
$$
\frac{\left(\frac{1}{2}\right)}{3}=\frac{1}{2} \times \frac{1}{3}=\frac{1}{6}
$$
and the denominator is 4 . Since the numerator is less than the denominator, the value of this fraction is less than 1 .
In option D the numerator is 1 and the denominator is
$$
\frac{2}{\left(\frac{3}{4}\right)}=2 \div \frac{3}{4}=2 \times \frac{4}{3}=\frac{8}{3} .
$$

Since the numerator is less than the denominator, the value of this fraction is less than 1 .
Finally, in option E the numerator is
$$
\frac{1}{\left(\frac{2}{3}\right)}=1 \div \frac{2}{3}=1 \times \frac{3}{2}=\frac{3}{2}
$$
and the denominator is 4. Again, the numerator is less than the denominator, and so the value of this fraction is also less than 1 .

It follows that option B has the largest value.
For inVeStigATION
7.1 Arrange the fractions given by the options in Question 7 in order of size.
7.2 The fractions given as options in Question 7 are all built up using 1,2,3 and 4 as single digits. There are other possibilities.
Which fraction built up using 1,2,3 and 4 as single digits has the largest value?
[The phrase "as single digits" means that the digits cannot be placed next to each other, so that, for example,
$$
\frac{432}{1}
$$
is not allowed. ]
7.3 Which fraction built up using 1, 2, 3, 4 and 5 as single digits has the largest value?
7.4 Simplify the fraction
$$
\frac{\left(\frac{\left(\frac{1}{2}\right)}{\left(\frac{3}{4}\right)}\right)}{\left(\frac{\left(\frac{5}{6}\right)}{\left(\frac{7}{8}\right)}\right)}
$$
by writing it in the form $\frac{p}{q}$, where $p$ and $q$ are positive integers which have no common factor other than 1 .