To solve this problem, we'll follow a step-by-step approach to determine the ratio of the area designated for decking to the area designated for grass.
1. Determine the fraction of the area that is grass:
We know that [tex]\(\frac{3}{7}\)[/tex] of the garden area is decking. Since the total area of the garden is 1 (or 100% of the garden), the fraction of the garden that is grass is given by:
[tex]\[
1 - \frac{3}{7}
\][/tex]
2. Calculate the fraction of the garden that is grass:
[tex]\[
1 - \frac{3}{7} = \frac{7}{7} - \frac{3}{7} = \frac{4}{7}
\][/tex]
So, [tex]\(\frac{4}{7}\)[/tex] of the garden is grass.
3. Express the ratio of decking to grass:
We now have the fractions for decking [tex]\(\left(\frac{3}{7}\right)\)[/tex] and for grass [tex]\(\left(\frac{4}{7}\right)\)[/tex]. The ratio of decking to grass is the fraction of the decking area divided by the fraction of the grass area:
[tex]\[
\frac{\frac{3}{7}}{\frac{4}{7}}
\][/tex]
4. Simplify the ratio:
To simplify this, we divide the numerator by the denominator:
[tex]\[
\frac{\frac{3}{7}}{\frac{4}{7}} = \frac{3}{7} \times \frac{7}{4} = \frac{3}{4}
\][/tex]
Therefore, the ratio of decking to grass in its simplest form is:
[tex]\[
\frac{3}{4}
\][/tex]
Thus, the ratio of the decking area to the grass area is [tex]\(\frac{3}{4}\)[/tex].
