Sure! Let's solve each part of this question step-by-step.
### (a) Finding the value of [tex]\( x \)[/tex].
(i) [tex]\( x:5 = 4:3 \)[/tex]
To solve this, we can set up a proportion:
[tex]\[
\frac{x}{5} = \frac{4}{3}
\][/tex]
Now, we use cross multiplication to find [tex]\( x \)[/tex]:
[tex]\[
x \cdot 3 = 4 \cdot 5
\][/tex]
So,
[tex]\[
3x = 20
\][/tex]
Finally, we solve for [tex]\( x \)[/tex] by dividing both sides by 3:
[tex]\[
x = \frac{20}{3} = 6.666666666666666
\][/tex]
So, the value of [tex]\( x \)[/tex] in part (i) is:
[tex]\[
x = 6.666666666666666
\][/tex]
(ii) [tex]\( 3:x = 2:5 \)[/tex]
Set up the proportion:
[tex]\[
\frac{3}{x} = \frac{2}{5}
\][/tex]
Using cross multiplication:
[tex]\[
3 \cdot 5 = 2 \cdot x
\][/tex]
So,
[tex]\[
15 = 2x
\][/tex]
Solving for [tex]\( x \)[/tex] by dividing both sides by 2:
[tex]\[
x = \frac{15}{2} = 7.5
\][/tex]
Thus, the value of [tex]\( x \)[/tex] in part (ii) is:
[tex]\[
x = 7.5
\][/tex]
(iii) [tex]\( 4:5 = x:\frac{1}{2} \)[/tex]
Set up the proportion:
[tex]\[
\frac{4}{5} = \frac{x}{\frac{1}{2}}
\][/tex]
Using cross multiplication:
[tex]\[
4 \cdot \frac{1}{2} = 5 \cdot x
\][/tex]
Which simplifies to:
[tex]\[
2 = 5x
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{2}{5} = 0.4
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] in part (iii) is:
[tex]\[
x = 0.4
\][/tex]
### Summary of Results:
- For (i) [tex]\( x = 6.666666666666666 \)[/tex]
- For (ii) [tex]\( x = 7.5 \)[/tex]
- For (iii) [tex]\( x = 0.4 \)[/tex]
