Let's solve the problem step by step.
### Part (i):
#### (a) Writing the present age of the elder sister in terms of x
Let the present age of the younger sister be [tex]\( x \)[/tex] years.
According to the problem, the difference in age between the elder and the younger sister is 5 years.
Hence, the present age of the elder sister can be expressed as:
[tex]\[ \text{Elder sister's age} = x + 5 \][/tex]
So, in terms of [tex]\( x \)[/tex], the present age of the elder sister is [tex]\( x + 5 \)[/tex] years.
#### (ii) Finding the value of [tex]\(x\)[/tex]
We are also given that the product of their ages is 84. Thus, we can set up the equation:
[tex]\[ x \times (x + 5) = 84 \][/tex]
Expanding this:
[tex]\[ x^2 + 5x - 84 = 0 \][/tex]
This is a quadratic equation and to solve this, we must find the roots. The roots of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] are found using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = -84 \)[/tex].
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-84)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-5 \pm \sqrt{25 + 336}}{2} \][/tex]
[tex]\[ x = \frac{-5 \pm \sqrt{361}}{2} \][/tex]
[tex]\[ x = \frac{-5 \pm 19}{2} \][/tex]
This gives us two possible solutions:
[tex]\[ x = \frac{-5 + 19}{2} = \frac{14}{2} = 7 \][/tex]
[tex]\[ x = \frac{-5 - 19}{2} = \frac{-24}{2} = -12 \][/tex]
Since age cannot be negative, we discard [tex]\( x = -12 \)[/tex].
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( 7 \)[/tex] years.
### Part (iii): Finding the present age of the elder sister
Now, we substitute [tex]\( x \)[/tex] back into the expression for the elder sister's age:
[tex]\[ \text{Elder sister's age} = x + 5 = 7 + 5 = 12 \text{ years} \][/tex]
Hence, the present age of the elder sister is 12 years.
### Part (iv): Finding their ages after 10 years
To find their ages after 10 years, we simply add 10 to their current ages.
- The younger sister's age after 10 years:
[tex]\[ 7 + 10 = 17 \text{ years} \][/tex]
- The elder sister's age after 10 years:
[tex]\[ 12 + 10 = 22 \text{ years} \][/tex]
### Summary:
- The present age of the younger sister is [tex]\( 7 \)[/tex] years.
- The present age of the elder sister is [tex]\( 12 \)[/tex] years.
- The age of the younger sister after 10 years will be [tex]\( 17 \)[/tex] years.
- The age of the elder sister after 10 years will be [tex]\( 22 \)[/tex] years.
