p and q are two numbers such that p>q .
When you subtract 5 from p and subtract 5 from q , the answers are in the ratio 9:1. 2. When you add 20 to p and add 20 to q , the answers are in the ratio 7:3. Find the ratio p:q . Give your answer in its simplest form.
The Correct Answer and Explanation is:
To solve for the ratio ( p:q ) based on the conditions provided, let’s denote the two numbers as ( p ) and ( q ).
Given Conditions:
- Condition 1: When 5 is subtracted from both numbers, the result is in the ratio ( 9:1 ).
[
\frac{p – 5}{q – 5} = \frac{9}{1}
]
This can be rewritten as:
[
p – 5 = 9(q – 5)
]
Expanding this gives:
[
p – 5 = 9q – 45 \implies p = 9q – 40 \quad \text{(Equation 1)}
] - Condition 2: When 20 is added to both numbers, the result is in the ratio ( 7:3 ).
[
\frac{p + 20}{q + 20} = \frac{7}{3}
]
This can be rewritten as:
[
3(p + 20) = 7(q + 20)
]
Expanding this gives:
[
3p + 60 = 7q + 140 \implies 3p = 7q + 80 \implies p = \frac{7q + 80}{3} \quad \text{(Equation 2)}
]
Solving the Equations
Now we have two equations:
- Equation 1: ( p = 9q – 40 )
- Equation 2: ( p = \frac{7q + 80}{3} )
Setting these equal to each other:
[
9q – 40 = \frac{7q + 80}{3}
]
To eliminate the fraction, multiply both sides by 3:
[
3(9q – 40) = 7q + 80
]
Expanding gives:
[
27q – 120 = 7q + 80
]
Now, isolate ( q ):
[
27q – 7q = 80 + 120 \implies 20q = 200 \implies q = 10
]
Finding ( p )
Substituting ( q = 10 ) back into Equation 1 to find ( p ):
[
p = 9(10) – 40 = 90 – 40 = 50
]
Finding the Ratio
Thus, we have:
- ( p = 50 )
- ( q = 10 )
Now, the ratio ( p:q ) is:
[
\frac{p}{q} = \frac{50}{10} = 5:1
]
Conclusion
The simplest form of the ratio ( p:q ) is ( 5:1 ). This result is consistent with the conditions provided, confirming the solution is correct. Thus, the answer is:
[
\boxed{5:1}
]
In summary, the method involved setting up equations based on the conditions given, solving these simultaneous equations, and deriving the values of ( p ) and ( q ) to determine their ratio.