What is the fourth proportional to the numbers 24, 120 and 22, that is, the number x such that 24 : 120 = 22 : x?

Introduction / Context:
This question asks for the fourth proportional to three numbers. The idea of fourth proportional is a classic concept in ratio and proportion, where we want to find a number x such that the proportion a : b = c : x holds. Here, a = 24, b = 120 and c = 22, and we must determine x.

Given Data / Assumptions:

• The first number is 24.• The second number is 120.• The third number is 22.• The fourth proportional x satisfies 24 : 120 = 22 : x.

Concept / Approach:
For numbers a, b, c and x, if a : b = c : x, then a / b = c / x. Cross multiplication gives a * x = b * c, and solving for x yields x = (b * c) / a. Applying this formula directly with the given numbers makes the calculation straightforward. We just need to perform a multiplication and a division accurately to obtain the fourth proportional.

Step-by-Step Solution:
Step 1: Write the proportion: 24 : 120 = 22 : x.Step 2: Convert the ratio into a fraction equation: 24 / 120 = 22 / x.Step 3: Cross multiply: 24 * x = 120 * 22.Step 4: Compute the right side: 120 * 22 = 120 * (20 + 2) = 120 * 20 + 120 * 2 = 2,400 + 240 = 2,640.Step 5: So we have 24 * x = 2,640.Step 6: Solve for x by dividing both sides by 24: x = 2,640 / 24.Step 7: Perform the division. First divide by 8: 2,640 / 8 = 330 and 24 / 8 = 3. So x = 330 / 3 = 110.Step 8: Therefore, the fourth proportional x is 110.

Verification / Alternative check:
To verify, check that 24 : 120 equals 22 : 110. Simplify 24 : 120 by dividing both terms by 24 to get 1 : 5. Now simplify 22 : 110 by dividing both by 22 to also get 1 : 5. Since both ratios simplify to the same value, the proportion 24 : 120 = 22 : 110 holds, confirming that x = 110 is correct.

Why Other Options Are Wrong:
• 120: This would give the ratio 22 : 120, which simplifies to 11 : 60 and does not equal 1 : 5.• 100: The ratio 22 : 100 simplifies to 11 : 50, which is different from 1 : 5.• 90: The ratio 22 : 90 simplifies to 11 : 45, again not matching 1 : 5.

Common Pitfalls:
Some learners reverse the numbers and mistakenly compute x = (a * c) / b instead of x = (b * c) / a. Others may incorrectly simplify the ratios or forget to reduce them to simplest form when checking their work. Arithmetic errors in multiplying 120 by 22 or dividing 2,640 by 24 can also cause trouble. Writing down the proportion carefully and using cross multiplication systematically helps avoid such mistakes.

Final Answer:
The fourth proportional to 24, 120 and 22 is 110.