In a consignment of electric bulbs, 4% were broken and from the remainder, 25% were found to be defective. If the total number of broken and defective bulbs is 112, then find the number of bulbs in the consignment.
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.
- Calculate the number of broken bulbs: Broken bulbs = 4% of $T$. In mathematical terms, this is $\frac{4}{100} \times T = 0.04T$.
- Calculate the number of remaining bulbs: After removing the broken bulbs, the remaining bulbs are $T - (\text{Broken bulbs})$. Remaining bulbs = $T - 0.04T = (1 - 0.04)T = 0.96T$.
- 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 = $0.24T$.
- 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 $112 = 0.04T + 0.24T$
- 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
| Concept | Explanation | Calculation Example (using total T) |
|---|---|---|
| Percentage of a Quantity | Finding a fraction of a total represented as a percentage. | $P\%$ of $T = \frac{P}{100} \times T$ |
| Finding Remainder | Subtracting a calculated part from the total. | Total - Part = Remainder |
| Percentage of Remainder | Calculating a percentage based on the value left after a previous calculation. | $Q\%$ of Remainder = $\frac{Q}{100} \times \text{Remainder}$ |
| Setting up Equations | Using algebraic equations to represent relationships between known and unknown quantities. | Sum of parts = Total (e.g., Broken + Defective = 112) |
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.
Important Questions from Quick Math
A is 120% of B and B is 65% of C. If the sum of A, B and C is 121.5, then the value of C - 2B + A is:
- A college hostel mess has provisions for 25 days for 350 boys. At the end of 10 days, when some boys were shifted to another hostel, it was found that now the provisions will last for 21 more days. How may boys were shifted to another hostel?
- Some students (only boys and girls) from different schools appeared for an Olympiad exam. 20% of the boys and 15% of the girls failed the exam. The number of boys who passed the exam was 70 more than that of the girls who passed the exam. A total of 90 students failed. Find the number of students that appeared for the exam.
The price of an item is reduced by 20%. As a result, customers can get 2 kg more of it for ₹360. Find the original price (in ₹) per kg of the item.
A tyre has 3 punctures. The first puncture alone would have made the tyre flat in 9 minutes, the second alone would have done it in 18 minutes, the third alone would have done it in 6 minutes. If the air leaks out at a constant rate, then how long (in minutes) does it take for all the punctures together to make it flat?
