Certainly! Let's tackle each part of the question step by step.
### Part (a): Simplify fully [tex]\(42: 18\)[/tex]
1. Find the Greatest Common Divisor (GCD):
To simplify the ratio [tex]\(42: 18\)[/tex], we first need to find the greatest common divisor (GCD) of 42 and 18. The GCD is the largest number that divides both 42 and 18 without leaving a remainder.
The GCD of 42 and 18 is 6.
2. Divide both terms by the GCD:
Now we will divide both numbers in the ratio by their GCD to simplify it.
[tex]\[
\frac{42}{6} : \frac{18}{6} = 7 : 3
\][/tex]
So, the fully simplified ratio [tex]\(42: 18\)[/tex] is [tex]\(7: 3\)[/tex].
### Part (b): Write the ratio [tex]\(9: 4\)[/tex] in the form [tex]\(n: 1\)[/tex]
1. Rewrite the ratio in the form [tex]\( n: 1 \)[/tex]:
To express the ratio [tex]\(9: 4\)[/tex] as [tex]\(n: 1\)[/tex], we'll divide both numbers by the second term, which is 4.
[tex]\[
\frac{9}{4} : \frac{4}{4} = 2.25 : 1
\][/tex]
So, the ratio [tex]\(9: 4\)[/tex] expressed in the form [tex]\(n: 1\)[/tex] is [tex]\(2.25: 1\)[/tex].
### Summary:
a) The simplified ratio [tex]\(42: 18\)[/tex] is [tex]\(7: 3\)[/tex].
b) The ratio [tex]\(9: 4\)[/tex] in the form [tex]\(n: 1\)[/tex] is [tex]\(2.25: 1\)[/tex].
