To find the two numbers given the LCM (Least Common Multiple) is 90, the HCF (Highest Common Factor) is 15, and their sum is 75, we can use the relationship between LCM and HCF.

Formula Relationship:

The relationship between two numbers $a$ and $b$, their LCM and HCF is given by:

$\text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b$

Step 1: Use the given information

We know:

• $\text{LCM}(a, b) = 90$
• $\text{HCF}(a, b) = 15$
• $a + b = 75$

From the formula, we can write:

$90 \times 15 = a \times b$

This simplifies to:

$a \times b = 1350$

Step 2: Use the sum of the numbers

We also know the sum of the two numbers is 75:

$a + b = 75$

We now have the system of two equations:

1. $a \times b = 1350$
2. $a + b = 75$

Step 3: Solve the system of equations

Let’s solve this using the quadratic equation. From the sum $a + b = 75$, we can express $b$ as:

$b = 75 - a$

Substitute this into the product equation:

$a \times (75 - a) = 1350$

This simplifies to:

$75a - a^2 = 1350$

Rearranging this gives:

$a^2 - 75a + 1350 = 0$

Step 4: Solve the quadratic equation

We can solve this quadratic equation using the quadratic formula:

$a = \frac{-(-75) \pm \sqrt{(-75)^2 - 4 \times 1 \times 1350}}{2 \times 1}$

Simplifying:

$a = \frac{75 \pm \sqrt{5625 - 5400}}{2}$ $a = \frac{75 \pm \sqrt{225}}{2}$ $a = \frac{75 \pm 15}{2}$

Thus, we have two possible solutions for $a$:

$a = \frac{75 + 15}{2} = 45 \quad \text{or} \quad a = \frac{75 - 15}{2} = 30$

Step 5: Find the values of $a$ and $b$

• If $a = 45$, then $b = 75 - 45 = 30$.
• If $a = 30$, then $b = 75 - 30 = 45$.

Thus, the two numbers are 30 and 45.

Step 6: Determine the greater number

The greater number is 45.

Final Answer:

The greater number is 45.

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Would you like further details on how to solve this or have any other questions?

Related Questions:

1. How do we derive the formula for LCM and HCF of two numbers?
2. How can the relationship between LCM and HCF be used in solving word problems?
3. What is the LCM and HCF of 30 and 45?
4. How can we solve a quadratic equation by completing the square?
5. How do we apply the LCM and HCF concept to more than two numbers?

Tip:

When solving problems involving LCM and HCF, remember to use the equation $\text{LCM} \times \text{HCF} = a \times b$, which is very useful when you know both the sum and product of the numbers.