Let's go through each part of the question step-by-step.
### Part (a)
#### (i) What should be added to these numbers to get a ratio of 5: 6?
We have two numbers: 6 and 9. We want to find a number [tex]\( x \)[/tex] such that when [tex]\( x \)[/tex] is added to both numbers, the ratio becomes 5:6.
Let:
[tex]\[ (6 + x) / (9 + x) = 5 / 6 \][/tex]
After solving this equation, the value of [tex]\( x \)[/tex] is found to be:
[tex]\[ x = 9 \][/tex]
Therefore, 9 should be added to the numbers 6 and 9 to get the ratio 5:6.
#### (ii) What should be subtracted from these numbers to get a ratio of 1: 2?
We have two numbers: 6 and 9. We want to find a number [tex]\( y \)[/tex] such that when [tex]\( y \)[/tex] is subtracted from both numbers, the ratio becomes 1:2.
Let:
[tex]\[ (6 - y) / (9 - y) = 1 / 2 \][/tex]
After solving this equation, the value of [tex]\( y \)[/tex] is found to be:
[tex]\[ y = 3 \][/tex]
Therefore, 3 should be subtracted from the numbers 6 and 9 to get the ratio 1:2.
### Part (b)
#### (i) If the smaller number is [tex]\( 4x \)[/tex], what is the bigger number?
We know the ratio of two numbers is 4:5. Let the smaller number be [tex]\( 4x \)[/tex].
Then the bigger number would be:
[tex]\[ 5x \][/tex]
#### (ii) If 6 is added to each number, write the new numbers so formed.
If 6 is added to each number:
- The smaller number becomes [tex]\( 4x + 6 \)[/tex]
- The bigger number becomes [tex]\( 5x + 6 \)[/tex]
#### (iii) If the ratio of the new numbers is 5:6, find the numbers.
We know that when 6 is added to each number, the ratio becomes 5:6.
So, we set up the equation:
[tex]\[ (4x + 6) / (5x + 6) = 5 / 6 \][/tex]
After solving this equation, there is no valid solution for [tex]\( x \)[/tex], hence there is no specific solution in terms of [tex]\( x \)[/tex].
### Last Part
Two numbers are in the ratio 7:5. When 10 is subtracted from each term, their ratio becomes 3:2. Find the numbers.
Let the two original numbers be [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
Given:
[tex]\[ a / b = 7 / 5 \][/tex]
[tex]\[ (a - 10) / (b - 10) = 3 / 2 \][/tex]
After solving these equations, we find:
[tex]\[ a = 70 \][/tex]
[tex]\[ b = 50 \][/tex]
So, the two numbers are 70 and 50.
