In an examination, 45% of all the students who appeared are boys and the rest are girls. If 60% of the boys and 70% of the girls passed, then what is the percentage of students who failed?
34.5
Understanding the Percentage Problem
This problem asks us to find the total percentage of students who failed an examination, given the distribution of boys and girls and their respective pass percentages.
Step-by-Step Solution
Let's break down the problem into smaller steps to find the total percentage of students who failed.
1. Determine the Percentage of Girls
The total percentage of students is 100%. If 45% are boys, the rest are girls.
Percentage of girls = Total percentage of students - Percentage of boys
Percentage of girls = $\text{100% - 45% = 55%}$
So, 55% of the students who appeared are girls.
2. Calculate the Percentage of Boys Who Failed
We are told that 60% of the boys passed. To find the percentage of boys who failed, we subtract the pass percentage from 100%.
Percentage of boys who failed = 100% - Percentage of boys who passed
Percentage of boys who failed = $\text{100% - 60% = 40%}$
This means 40% of the boys failed the examination.
3. Calculate the Percentage of Girls Who Failed
Similarly, 70% of the girls passed. The percentage of girls who failed is:
Percentage of girls who failed = 100% - Percentage of girls who passed
Percentage of girls who failed = $\text{100% - 70% = 30%}$
So, 30% of the girls failed the examination.
4. Calculate the Percentage of Total Students Who Are Failed Boys
We know that 45% of the total students are boys, and 40% of these boys failed. To find what percentage of the *total* students are failed boys, we calculate 40% of 45%.
Percentage of total students who are failed boys = Percentage of boys $\times$ Percentage of boys who failed
Percentage of total students who are failed boys = $\text{45% of 40% = } \frac{45}{100} \times \frac{40}{100}$
Percentage of total students who are failed boys = $\frac{45 \times 40}{100 \times 100} = \frac{1800}{10000} = 0.18$
Converting this back to a percentage of the total students: $\text{0.18} \times \text{100% = 18%}$.
So, 18% of the total students are boys who failed.
5. Calculate the Percentage of Total Students Who Are Failed Girls
We know that 55% of the total students are girls, and 30% of these girls failed. To find what percentage of the *total* students are failed girls, we calculate 30% of 55%.
Percentage of total students who are failed girls = Percentage of girls $\times$ Percentage of girls who failed
Percentage of total students who are failed girls = $\text{55% of 30% = } \frac{55}{100} \times \frac{30}{100}$
Percentage of total students who are failed girls = $\frac{55 \times 30}{100 \times 100} = \frac{1650}{10000} = 0.165$
Converting this back to a percentage of the total students: $\text{0.165} \times \text{100% = 16.5%}$.
So, 16.5% of the total students are girls who failed.
6. Calculate the Total Percentage of Students Who Failed
The total percentage of students who failed is the sum of the percentage of failed boys (as a percentage of total students) and the percentage of failed girls (as a percentage of total students).
Total percentage of students who failed = Percentage of total students who are failed boys + Percentage of total students who are failed girls
Total percentage of students who failed = $\text{18% + 16.5% = 34.5%}$
Therefore, the percentage of students who failed is 34.5%.
Here is a summary of the calculations:
| Category | Percentage of Total Students | Pass Percentage (within category) | Fail Percentage (within category) | Percentage of Total Students Who Failed (Category % × Fail %) |
|---|---|---|---|---|
| Boys | 45% | 60% | $\text{100% - 60% = 40%}$ | $\text{45% of 40% = 18%}$ |
| Girls | 55% | 70% | $\text{100% - 70% = 30%}$ | $\text{55% of 30% = 16.5%}$ |
Total Failed Percentage = Percentage of total students who are failed boys + Percentage of total students who are failed girls
Total Failed Percentage = $\text{18% + 16.5% = 34.5%}$
Revision Table: Key Concepts in Percentage Problems
| Concept | Description | Formula/Method |
|---|---|---|
| Percentage | A fraction of 100. Represented by the symbol %. | $\text{Percentage = } \frac{\text{Part}}{\text{Whole}} \times \text{100%}$ |
| Finding a Percentage of a Quantity | Calculating a portion of a given number based on a percentage. | $\text{Percentage of Quantity = } \frac{\text{Given Percentage}}{100} \times \text{Quantity}$ |
| Complementary Percentages | If a percentage represents a part, the remaining percentage represents the rest (up to 100%). | If P% of something is X, then (100-P)% is the rest. |
| Combining Percentages (Weighted Average) | When percentages are from different subgroups, they must be converted to percentages of the total group before adding. | $\text{Overall % = (Group1 % of Total) } \times \text{ (Result % in Group1) + (Group2 % of Total) } \times \text{ (Result % in Group2)}$ |
Additional Information: Solving Percentage Problems
When tackling percentage problems involving different groups, it's crucial to distinguish between percentages calculated within a subgroup and percentages calculated with respect to the total. In this problem, 60% is the pass percentage *of the boys*, not 60% of the *total* students. Similarly, 70% is the pass percentage *of the girls*, not 70% of the *total* students.
To find the percentage of failed students out of the total, you must first find the number (or percentage) of failed students in each group and then express this number (or percentage) as a proportion of the total number of students. Summing these proportions gives the total percentage of students who failed.
Using 100 as a base value for the total number of students can often simplify percentage calculations, as percentages directly translate to numbers. If there were 100 students:
- Boys = 45
- Girls = 55
- Boys who failed = 40% of 45 = $\frac{40}{100} \times 45 = 0.4 \times 45 = 18$
- Girls who failed = 30% of 55 = $\frac{30}{100} \times 55 = 0.3 \times 55 = 16.5$
- Total students who failed = $18 + 16.5 = 34.5$
Since we assumed a total of 100 students, the total number of failed students (34.5) is also the total percentage of students who failed (34.5%).
Similar Questions
Chamanlal, Arshad and Jagit Singh contested an election. All the votes polled were valid. Arshad got 35% of the total votes. For every 35 votes Chamanlal got 14 votes. The winner got 4950 more votes than the person who received the least number of votes. Find the total number of votes polled.
The sum of two numbers is 680. If the bigger number is decreased by 15% and the smaller number is increased by 15%, then the resultant numbers are equal. Find the smaller number.
A number is decreased by 20% to get another number. The number so obtained is increased by 200% to get a third number. The difference of the third number and the original number is what percentage more or less than the difference of second and the third number?
In the first year the population of a town decreased by 5% due to the Corona virus first wave. In the next year it decreased again by 5% due to the second wave and in the third year it increased by 5%. At the end of the third year the population was 9,47,625. What was the population at the beginning of the first year?
Radha saves 25% of her income. If her expenditure increases by 20% and her income increases by 29%, then her savings increase by;
In an examination, 92% of the students passed and 480 students failed. If so, how many students appeared in the examination?
If the sum of 40% of a number and 30% of the same number is 70, then the number is:
- \(\frac{{25\% \;of\;(50\% \;of\;30\% \;of\;150)}}{{40\% \;of\;2250}}\) is equal to:
By mistake, the reciprocal of a positive fraction got typed in place of itself, and thereby its value got reduced by \(\frac{175}{4}\%\) . What was the value of the fraction?
Rohit's income is Rs. 32000. If his expenses is 30 percent of total income, then what will be the saving of Rohit ?
Important Questions from Percentage
In an examination, 25% of the candidates failed in Mathematics and 12% failed in English. If 10% of the candidates failed in both the subjects and 292 candidates passed in both the subjects, which one of the following is the number of total candidates appeared in the examination?
What is the value of 9% of 5500 + 2.4% of 1100 - 40% of 1600?
Population of a village is 7960 in which 4660 are female. If in that village 60% are literate in which 70% female are literate, then what is the number of literate male ?
In an examination, Ram obtained 20 % more than Ashok but 10% less than Rajesh. If the marks obtained by Ashok is 1080. Then the Percentage marks obtained by Rajesh if the full marks is 2000 ;
A student appeared in 6 papers. The maximum marks are the same for each paper. His marks in these papers are in the proportion of 5 : 6 : 7 : 8 : 9 : 10. Overall he scored 60%. In how many number of papers did he score less than 60% of the maximum marks?
