Solving the Electric Bulb Consignment Problem

This problem involves calculating the total number of items in a consignment based on given percentages of broken and defective items and the total count of these items.

Let's break down the problem step by step to find the total number of bulbs in the consignment.

Understanding the Problem Percentages

We are given:

• 4% of the total bulbs were broken.
• 25% of the remainder (after removing broken bulbs) were defective.
• The total number of broken and defective bulbs is 112.

Step-by-Step Solution to Find Total Bulbs

Let $T$ represent the total number of bulbs in the consignment.

1. Calculate the number of broken bulbs: Broken bulbs = 4% of $T$. In mathematical terms, this is $\frac{4}{100} \times T = 0.04T$.
2. Calculate the number of remaining bulbs: After removing the broken bulbs, the remaining bulbs are $T - (\text{Broken bulbs})$. Remaining bulbs = $T - 0.04T = (1 - 0.04)T = 0.96T$.
3. Calculate the number of defective bulbs: Defective bulbs = 25% of the remainder. Defective bulbs = 25% of $0.96T$. In mathematical terms, this is $\frac{25}{100} \times 0.96T = 0.25 \times 0.96T$. Calculating the value: $0.25 \times 0.96 = \frac{1}{4} \times \frac{96}{100} = \frac{96}{400} = \frac{24}{100} = 0.24$. So, defective bulbs = $0.24T$.
4. Set up an equation based on the total broken and defective bulbs: The total number of broken and defective bulbs is given as 112. Total broken and defective bulbs = Broken bulbs + Defective bulbs $112 = 0.04T + 0.24T$
5. Solve the equation for $T$: Combine the terms with $T$: $112 = (0.04 + 0.24)T$ $112 = 0.28T$ To find $T$, divide 112 by 0.28: $T = \frac{112}{0.28}$ To simplify the division, multiply the numerator and denominator by 100 to remove the decimal: $T = \frac{112 \times 100}{0.28 \times 100} = \frac{11200}{28}$ Now, perform the division: $11200 \div 28$ We know that $28 \times 4 = 112$. So, $28 \times 400 = 11200$. $T = 400$.

Therefore, the total number of bulbs in the consignment is 400.

Summary of Bulb Count

Let's verify the numbers with $T = 400$:

• Total bulbs = 400
• Broken bulbs = 4% of 400 = $0.04 \times 400 = 16$
• Remaining bulbs = $400 - 16 = 384$
• Defective bulbs = 25% of remainder = 25% of 384 = $0.25 \times 384 = \frac{1}{4} \times 384 = 96$
• Total broken and defective bulbs = Broken bulbs + Defective bulbs = $16 + 96 = 112$.

This matches the information given in the problem, confirming our calculation is correct.

Revision Table: Key Concepts in Percentage Problems

Additional Information: Percentage Calculations Explained

Percentages are a way to express a part of a whole as a fraction of 100. The term "percent" means "out of 100".

• Converting Percentage to Decimal: To convert a percentage (like $P\%$) to a decimal, divide by 100. For example, $4\% = \frac{4}{100} = 0.04$.
• Calculating Percentage of a Number: To find $P\%$ of a number $N$, you calculate $\frac{P}{100} \times N$. In this problem, we used this to find the number of broken and defective bulbs.
• Working with Remainders: When a percentage is taken "from the remainder," it's crucial to first subtract the amount already accounted for from the original total before calculating the second percentage. In our problem, 25% was taken from the bulbs that were *not* broken.
• Solving for the Unknown Total: Often, percentage problems give you the values of parts and ask for the original total. This involves setting up an equation where the unknown total is a variable (like $T$) and solving for it. This typically involves combining terms and using inverse operations (like division) to isolate the variable.

Understanding how to convert percentages to decimals or fractions and how to correctly identify the base amount (the "whole") for each percentage calculation is key to solving these types of problems accurately.