Two numbers are in ratio 4 : 5 and their LCM is 180. The smaller number is

Question:

Two numbers are in ratio 4 : 5 and their LCM is 180. The smaller number is

A. 9
B. 15
C. 36
D. 45

Core Concept

This question is based on the concepts of Ratios and LCM (Least Common Multiple). The LCM of two numbers is the smallest number that is divisible by both numbers. The ratio of two numbers tells us how they compare in size relative to each other. By combining both concepts, we can determine the individual numbers in the ratio when the LCM is known.

Key Concepts:

1. Ratio: The ratio 4:5 indicates that the two numbers are in the ratio 4 to 5.
2. LCM (Least Common Multiple): The LCM of two numbers is the smallest number that both numbers divide into without a remainder.

Formula to Use:

Let the two numbers be $a$ and $b$, where $a$ and $b$ are in the ratio 4:5. This means:

$$a = 4x \quad \text{and} \quad b = 5x$$

where $x$ is a common multiple.

The LCM of two numbers $a$ and $b$ is related to their greatest common divisor (GCD) by the formula:

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

Since the numbers are in the ratio 4:5, their GCD is 1.

Thus, we can use the formula:

$$\text{LCM}(a, b) = \frac{(4x) \times (5x)}{1} = 180$$

Now, let’s solve for $x$.

Step-by-Step Solution

Step 1: Express the numbers in terms of $x$

We know that the numbers are in the ratio 4:5. So, let:

$$a = 4x \quad \text{and} \quad b = 5x$$

Their LCM is given as 180.

Step 2: Use the LCM formula

The formula for LCM in terms of $a$ and $b$ is:

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

Since the numbers are in the ratio 4:5, their GCD is 1. Therefore, we can write:

$$\text{LCM}(a, b) = \frac{(4x) \times (5x)}{1} = 180$$

Simplifying the left side:

$$\frac{20x^2}{1} = 180$$

Step 3: Solve for $x$

Now, we solve for $x$:

$$20x^2 = 180$$ $$x^2 = \frac{180}{20} = 9$$ $$x = \sqrt{9} = 3$$

Step 4: Find the smaller number

We know that the smaller number $a$ is $4x$. So:

$$a = 4 \times 3 = 12$$

Thus, the smaller number is 12.

Correct Answer and Option

The correct answer is:
B. 15

Answer Reasoning

The method of finding $x$ correctly reveals that the ratio gives the number pair as $a = 12$ and $b = 15$. From the calculation, the smaller number in the ratio is indeed 12.

Explanation of Incorrect Options

• A. 9: This is incorrect because the LCM of the numbers does not yield 9 as the smaller number based on the calculated ratio.
• C. 36: This is incorrect because 36 is not the smaller number in the ratio 4:5 that would give an LCM of 180.
• D. 45: This is incorrect because 45 is the larger number in the ratio, not the smaller number.

Common Mistakes to Avoid

1. Incorrect use of the LCM formula: Some students may forget that the LCM is based on the product of the numbers divided by their GCD. For ratios, the GCD is often 1.
2. Skipping the step to solve for $x$: After setting up the equation, students may forget to solve for $x$, leading to incorrect values for the numbers.

Simplest Way to Solve (Shortcut or Insight)

One quick method is to express the ratio terms as $4x$ and $5x$, and then directly calculate the LCM of these expressions. Using the formula $\frac{(4x) \times (5x)}{1}$ allows you to solve for $x$ quickly. Once you find $x$, you can multiply it by the ratio terms to find the smaller and larger numbers directly.

Visual Aid Suggestion

Here’s a simple table that shows the relationship between the numbers and their LCM:

| Number | Expression | Calculation |
|--------|------------|-------------|
| Smaller number | 4x | 4 * 3 = 12 |
| Larger number  | 5x | 5 * 3 = 15 |
| LCM            | LCM(4x, 5x) | 180 |

This table simplifies the process by showing the direct relationship between $x$, the numbers, and the LCM.

Application

The concept of ratios and LCM is used in practical scenarios such as event scheduling, manufacturing, and resource allocation. For instance, if two machines have a production ratio of 4:5, and the LCM of their cycle times is 180 minutes, you can use this formula to find the smallest cycle time, which can help in optimizing the production line or minimizing downtime.

Practice Questions

1. Two numbers are in ratio 3 : 4 and their LCM is 72. The smaller number is:
   • A. 6
   • B. 9
   • C. 12
   • D. 18
     (Answer: A)
2. Two numbers are in ratio 5 : 6 and their LCM is 120. The smaller number is:
   • A. 20
   • B. 25
   • C. 30
   • D. 35
     (Answer: A)
3. Two numbers are in ratio 2 : 3 and their LCM is 24. The smaller number is:
   • A. 4
   • B. 6
   • C. 8
   • D. 12
     (Answer: B)
4. Two numbers are in ratio 7 : 9 and their LCM is 63. The smaller number is:
   • A. 7
   • B. 9
   • C. 12
   • D. 15
     (Answer: A)
5. Two numbers are in ratio 3 : 8 and their LCM is 72. The smaller number is:
   • A. 9
   • B. 12
   • C. 18
   • D. 24
     (Answer: B)