There are 350 boys in the first three standard. The ratio of the number of boys in first and second standards is 2 : 3, while the ratio of boys in second and third standard is 4 : 5. What is the total number of boys in first and third standards?
230
Understanding the Ratio Problem
The question provides information about the number of boys in the first three standards. We are given the total number of boys across these three standards and the ratios of boys between consecutive standards. Our goal is to find the combined number of boys in the first and third standards.
Given Information:
- Total number of boys in standard 1, standard 2, and standard 3 = 350
- Ratio of boys in standard 1 to standard 2 = 2 : 3
- Ratio of boys in standard 2 to standard 3 = 4 : 5
Finding the Combined Ratio
To find the total number of boys in standard 1 and standard 3, we first need to determine the ratio of boys across all three standards: standard 1 : standard 2 : standard 3. We are given two separate ratios:
- Std 1 : Std 2 = 2 : 3
- Std 2 : Std 3 = 4 : 5
Notice that 'standard 2' is common in both ratios, but its ratio value is different (3 in the first ratio and 4 in the second). To combine these ratios, we need to make the 'standard 2' part equal in both ratios. We can do this by finding the Least Common Multiple (LCM) of the two values for standard 2, which are 3 and 4.
The LCM of 3 and 4 is 12.
Now, we adjust each ratio so that the standard 2 part becomes 12:
- For Std 1 : Std 2 = 2 : 3, multiply both parts by the factor needed to make 3 equal to 12. This factor is \(12 \div 3 = 4\). \( (2 \times 4) : (3 \times 4) = 8 : 12 \) So, Std 1 : Std 2 = 8 : 12.
- For Std 2 : Std 3 = 4 : 5, multiply both parts by the factor needed to make 4 equal to 12. This factor is \(12 \div 4 = 3\). \( (4 \times 3) : (5 \times 3) = 12 : 15 \) So, Std 2 : Std 3 = 12 : 15.
Now that the ratio value for standard 2 is the same (12) in both adjusted ratios, we can combine them to get the ratio for all three standards:
Std 1 : Std 2 : Std 3 = 8 : 12 : 15.
Calculating the Number of Boys in Each Standard
The combined ratio 8 : 12 : 15 means that the total number of boys is divided into \(8 + 12 + 15\) parts. Total parts in the ratio = \(8 + 12 + 15 = 35\) parts.
We know the total number of boys is 350. So, these 35 parts represent 350 boys.
Value of one ratio part = \( \frac{\text{Total boys}}{\text{Total ratio parts}} \)
\( \text{Value of one part} = \frac{350}{35} = 10 \)
Each part of the ratio represents 10 boys.
Now we can find the number of boys in each standard:
- Number of boys in Standard 1 = 8 parts \(\times\) 10 boys/part = 80 boys.
- Number of boys in Standard 2 = 12 parts \(\times\) 10 boys/part = 120 boys.
- Number of boys in Standard 3 = 15 parts \(\times\) 10 boys/part = 150 boys.
Let's check if the total matches the given information: \(80 + 120 + 150 = 350\). This matches the total number of boys given in the question.
Finding the Total Number of Boys in First and Third Standards
The question asks for the total number of boys in the first and third standards. We have calculated the number of boys in each standard:
- Boys in Standard 1 = 80
- Boys in Standard 3 = 150
Total boys in first and third standards = Boys in Standard 1 + Boys in Standard 3
\( \text{Total} = 80 + 150 = 230 \)
Summary of Steps:
- Write down the given ratios.
- Find the LCM of the common part in the ratios (Standard 2).
- Adjust the individual ratios to match the LCM.
- Combine the adjusted ratios to get a single ratio for all standards.
- Calculate the total number of parts in the combined ratio.
- Divide the total number of boys by the total ratio parts to find the value of one part.
- Multiply the value of one part by the respective ratio part to find the number of boys in each standard.
- Add the number of boys in the required standards (Standard 1 and Standard 3).
| Standard | Ratio Part | Number of Boys (Ratio Part × 10) |
|---|---|---|
| Standard 1 | 8 | 80 |
| Standard 2 | 12 | 120 |
| Standard 3 | 15 | 150 |
| Total | 35 | 350 |
The total number of boys in the first and third standards is \(80 + 150 = 230\).
Revision Table: Ratio Problem Solving
| Concept | Description | Application in this Problem |
|---|---|---|
| Ratio Combination | Combining two or more ratios with a common element by making the value of the common element equal using LCM. | Combining Std 1:Std 2 (2:3) and Std 2:Std 3 (4:5) via Std 2 using LCM of 3 and 4 (which is 12) to get Std 1:Std 2:Std 3 as 8:12:15. |
| Ratio to Quantity | Converting ratio parts into actual quantities when the total quantity is known. Total Quantity / Total Ratio Parts = Value of one part. | Total boys (350) divided by total ratio parts (35) gives 10 boys per part. |
| Calculating Parts | Finding the number of items for specific parts of the ratio by multiplying the value of one part by the ratio part. | Boys in Std 1 = 8 parts * 10 = 80. Boys in Std 3 = 15 parts * 10 = 150. |
Additional Information: Ratio Concepts
Ratios are used to compare quantities of the same kind. They show how much of one quantity there is compared to another quantity. A ratio can be written using a colon (e.g., 2:3), as a fraction (e.g., \(\frac{2}{3}\)), or with the word 'to' (e.g., 2 to 3).
Key Concepts:
- Simplest Form: Ratios are often expressed in their simplest form, just like fractions. For example, 4:6 is simplified to 2:3 by dividing both numbers by their greatest common divisor, 2.
- Proportion: A proportion is an equation that states that two ratios are equivalent. For example, 2:3 = 4:6 is a proportion.
- Direct Proportion: Two quantities are in direct proportion if they increase or decrease at the same rate. If quantity A is directly proportional to quantity B, then \(A/B\) is constant.
- Inverse Proportion: Two quantities are in inverse proportion if an increase in one quantity leads to a decrease in the other quantity at a constant rate. If quantity A is inversely proportional to quantity B, then \(A \times B\) is constant.
In this problem, we used the concept of combining ratios, which is fundamental in solving problems where quantities are related through multiple ratios involving a common link.
Similar Questions
What number must be subtracted from both the numerator and denominator of the fraction 27/35 so that it becomes 2/3?
If a, b, c, d, e and f satisfy
2a = 3b = 6c = 9d = 12e = 18f, then what is the value of (a + b)/(c + d + e + f)?
To maintain 8 cows for 60 days, a milkman spends Rs. 6400. To maintain 5 cows for n days, he spends Rs. 4800. What is the value of n?
4 goats or 6 sheep can graze a field in 50 days. 2 goats and 3 sheep will graze it in
The annual incomes of two persons are in the ratio 9 ∶ 7 and their expenses are in the ratio 4 ∶ 3. If each of them saves Rs. 2000 per year, what is the difference in their annual incomes?
In a class of 49 students, the ratio of girls to boys is 4 ∶ 3. If 4 girls leave the class, the ratio of girls to boys would be
When a ball is allowed to fall, the time it takes to fall any distance varies as the square root of the distance and it takes 4 seconds to fall 78.40 m. How long would it take to fall 122.50 m?
Incomes of Mahesh and Kamal are in the ratio 1 : 2 and their expenses are in the ratio 1 : 3. Which one of the following is correct?
In an office, one-third of the workers are women, half of the women are married, and one-third of the married women have children. If three-fourth of the men are married and one-third of the married men have children, then what is the ratio of married women to married men?
The incomes of A, B and C are in the ratio 7 ∶ 9 ∶ 12 and their expenditures are in the ratio 8 ∶ 9 ∶ 15. If A's saving is one-fourth of his income, then the ratio of savings of A, B and C is
Important Questions from Simple Ratios
If the value of \(\frac{{p - q}}{q} = \frac{4}{3}\) .find p : q
Half of the villagers of a certain village have their own houses. One-fifth of the villagers cultivate paddy. One-third of the villagers are literate. Four-fifth of the villagers are under 25 years of age. Which one of the following statements is certainly correct ?
The two numbers are in the ratio of 9 ∶ 7 and the difference between of these two number is 6000. What is the sum of the two numbers?
A certain number is added to each of a pair of numbers which are in the ratio 4 ∶ 5. The sum of the resulting numbers is 39 and their ratio (taken in the same order as mentioned above) is 6 ∶ 7. What is the number added?
If (p + q) ∶ (q + r) ∶ (r + p) = 5 ∶ 6 ∶ 7 and p + q + r = 18, find the value of p ∶ q ∶ r
