If 35% of a number is 24 less than 50% of that number, what is the number?

Introduction / Context:
This is an algebraic percentage problem that involves comparing two different percentages of the same unknown number. The statement says that 35% of a number is 24 less than 50% of that number. From this information we need to determine the original number. Such questions train the ability to translate verbal percentage relations into equations and solve them efficiently.

Given Data / Assumptions:

Let the unknown number be N.

35% of N is 24 less than 50% of N.

In equation form, 35% of N = 50% of N minus 24.

We are required to find the value of N.

Concept / Approach:
Convert the percentages into their decimal forms and then set up an equation. Specifically, 35% of N is 0.35N and 50% of N is 0.50N. The statement that one is 24 less than the other can be written as 0.35N = 0.50N - 24. Solving this linear equation in N involves bringing like terms together and isolating N. This straightforward method is standard in many percentage comparison questions.

Step-by-Step Solution:
Let the number be N.35% of N = 0.35N.50% of N = 0.50N.Given that 35% of N is 24 less than 50% of N.So, 0.35N = 0.50N - 24.Bring terms involving N to one side: 0.50N - 0.35N = 24.0.15N = 24.Hence N = 24 / 0.15.Compute 24 / 0.15 = 2400 / 15 = 160.Therefore, the number is 160.

Verification / Alternative check:
Check by substituting N = 160 back into the original condition. 35% of 160 = 0.35 * 160 = 56. 50% of 160 = 0.50 * 160 = 80. Now compute the difference: 80 - 56 = 24. This matches the condition that 35% of the number is 24 less than 50% of the number. Thus, 160 is correct.

Why Other Options Are Wrong:

80 would give 35% of 80 = 28 and 50% of 80 = 40, with a difference of 12, not 24.

100 would give 35 and 50, with a difference of 15, not 24.

120 would give 42 and 60, with a difference of 18, not 24.

140 would give 49 and 70, with a difference of 21, also not equal to 24.

Common Pitfalls:
One common mistake is to interpret 35% as 0.35 and 50% as 0.5 correctly, but then to add or subtract the percentages themselves instead of forming the equation with N. Another error is in handling the algebra, for example writing 0.35N - 0.50N = 24, which leads to a negative coefficient. Misplacing the decimal points when dividing 24 by 0.15 can also cause errors. Using fraction forms, such as 35 / 100 and 50 / 100, can help if decimals feel confusing.

Final Answer:
The number that satisfies the given percentage condition is 160.