Two numbers are in ratio 7:9. If their H.C.F is 5, find the two numbers.14,1821,3135,4512,1428,36

1. Break Down the Problem

We need to find two numbers that are in the ratio $7:9$ and have a highest common factor (H.C.F) of $5$.

2. Relevant Concepts

If two numbers $a$ and $b$ are in the ratio $m:n$, we can express them as:

• $a = 7k$
• $b = 9k$

where $k$ is a common multiplier. Given that the H.C.F. of the two numbers is $5$, we can set up the equation based on the relation with the H.C.F:

• $H.C.F(a, b) = k \cdot H.C.F(7, 9) = k$

3. Analysis and Detail

Since $H.C.F(7, 9) = 1$, we have: $H.C.F(a, b) = k \cdot 1 = k$ Given $H.C.F(a, b) = 5$, we can now set: $k = 5$

Now substituting back to find $a$ and $b$:

• $a = 7k = 7 \times 5 = 35$
• $b = 9k = 9 \times 5 = 45$

4. Verify and Summarize

The two numbers are $35$ and $45$. We can check that:

• The ratio of $35:45$ simplifies to $7:9$.
• The highest common factor of $35$ and $45$ indeed confirms as $5$.

Final Answer

The two numbers are $35$ and $45$.