Sure, let's solve the given problems step by step.
### Part (i) What should be added to these numbers to get a ratio of 5:67
Let's assume that the number to be added is [tex]\( x \)[/tex].
Given two numbers are 6 and 9.
We want the new numbers (6 + [tex]\( x \)[/tex]) and (9 + [tex]\( x \)[/tex]) to be in the ratio of 5:67.
This can be represented by the equation:
[tex]\[
\frac{6 + x}{9 + x} = \frac{5}{67}
\][/tex]
Cross-multiplying to eliminate the fraction gives:
[tex]\[
67(6 + x) = 5(9 + x)
\][/tex]
Expanding both sides:
[tex]\[
67 \cdot 6 + 67 \cdot x = 5 \cdot 9 + 5 \cdot x
\][/tex]
[tex]\[
402 + 67x = 45 + 5x
\][/tex]
To isolate [tex]\( x \)[/tex], move all terms involving [tex]\( x \)[/tex] to one side and constants to the other side:
[tex]\[
67x - 5x = 45 - 402
\][/tex]
[tex]\[
62x = -357
\][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-357}{62}
\][/tex]
Simplifying the fraction:
[tex]\[
x = -5.75806451612903
\][/tex]
So, the number that should be added to both numbers to get a ratio of 5:67 is approximately [tex]\(-5.758\)[/tex].
### Part (ii) What should be subtracted from these numbers to get a ratio of 1:2
Let's assume that the number to be subtracted is [tex]\( y \)[/tex].
Given two numbers are 6 and 9.
We want the new numbers (6 - [tex]\( y \)[/tex]) and (9 - [tex]\( y \)[/tex]) to be in the ratio of 1:2.
This can be represented by the equation:
[tex]\[
\frac{6 - y}{9 - y} = \frac{1}{2}
\][/tex]
Cross-multiplying to eliminate the fraction gives:
[tex]\[
2(6 - y) = 1(9 - y)
\][/tex]
Expanding both sides:
[tex]\[
2 \cdot 6 - 2 \cdot y = 9 - y
\][/tex]
[tex]\[
12 - 2y = 9 - y
\][/tex]
To isolate [tex]\( y \)[/tex], move all terms involving [tex]\( y \)[/tex] to one side and constants to the other side:
[tex]\[
12 - 9 = -y + 2y
\][/tex]
[tex]\[
3 = y
\][/tex]
So, the number that should be subtracted to make the ratio 1:2 is:
[tex]\[
y = 3
\][/tex]
### Summary
(i) To get a ratio of 5:67, you need to add approximately [tex]\(-5.758\)[/tex] to both numbers.
(ii) To get a ratio of 1:2, you need to subtract 3 from both numbers.
