Certainly! Let's tackle the question step-by-step, focusing on part (c) first, as it provides a foundation for understanding the problem.
### Part (c)
Given:
- The ratio of two numbers is 7:5.
- When 10 is subtracted from each term, the new ratio becomes 3:2.
Step-by-Step Solution:
1. Define the numbers:
Using the given information, let the two numbers be [tex]\( 7x \)[/tex] and [tex]\( 5x \)[/tex].
2. Set up the equation:
When 10 is subtracted from each number, their new ratio becomes 3:2. Thus, we have:
[tex]\[
\frac{7x - 10}{5x - 10} = \frac{3}{2}
\][/tex]
3. Cross-multiply to eliminate the fraction:
[tex]\[
2(7x - 10) = 3(5x - 10)
\][/tex]
4. Expand both sides:
[tex]\[
14x - 20 = 15x - 30
\][/tex]
5. Isolate [tex]\( x \)[/tex]:
- Subtract [tex]\( 14x \)[/tex] from both sides:
[tex]\[
-20 = x - 30
\][/tex]
- Add 30 to both sides:
[tex]\[
10 = x
\][/tex]
Thus, [tex]\( x = 10 \)[/tex].
6. Calculate the actual numbers:
- First number: [tex]\( 7x = 7 \times 10 = 70 \)[/tex]
- Second number: [tex]\( 5x = 5 \times 10 = 50 \)[/tex]
Therefore, the two numbers are 70 and 50.
Moving on to part (ii):
### Part (ii)
Given:
- These two numbers now have a new ratio of 5:6.
Step-by-Step Solution:
1. Numbers identified in part (c):
From part (c), the numbers are 70 and 50.
2. Define the new numbers:
Let these new changed numbers be [tex]\( 5y \)[/tex] and [tex]\( 6y \)[/tex].
3. Set up the proportion:
[tex]\[
\frac{70}{50} = \frac{5y}{6y}
\][/tex]
4. Simplify the left side:
[tex]\[
\frac{7}{5} = \frac{5y}{6y}
\][/tex]
5. Cross-multiply to solve for [tex]\( y \)[/tex]:
[tex]\[
7 \cdot 6y = 5 \cdot 5y
\][/tex]
[tex]\[
42y = 25y
\][/tex]
6. Isolate and solve for [tex]\( y \)[/tex]:
(Note: This step seems to be inconsistent with the given context. You may need to adjust numbers but following procedure)
We get both the sides equal. The numbers should be consistent with the basic ratio properties.
At this point, revisit the initially obtained ratio:
- Regarding adjustments needed, the proportion consistency is seemed to be expressed consistently.
### Part (a)
Given:
- The ratio of the present age of a father and his son is 3:1.
Step-by-Step Solution:
Given the ratio:
- Let the father's age be [tex]\( 3z \)[/tex] and the son's age be [tex]\( z \)[/tex].
We don't have additional details such as their sums or increments, let's denote ages as above simple algebraically according to their ratio.
Thus, the simplified present age ratio representation is:
- Father's age: [tex]\( 3z \)[/tex]
- Son's age: [tex]\( z \)[/tex]
Further details for age specifics might depend on further piece data beyond the ratio context.
