In a box, the ratio of number of green stones to black stones is 5 ∶ 8. If there are 24 black stones, then how many total stones are in the box?
39
Understanding the Ratio of Stones
This problem asks us to find the total number of stones in a box, given the ratio of green stones to black stones and the actual count of black stones. The key concept here is understanding and applying ratios to find unknown quantities.
Analyzing the Given Information
- Ratio of green stones to black stones = 5 : 8
- Number of black stones = 24
We need to find the total number of stones in the box.
Solving the Ratio Problem Step-by-Step
The ratio 5 : 8 means that for every 5 green stones, there are 8 black stones. We can represent the number of green stones as a multiple of 5 and the number of black stones as the same multiple of 8. Let this common multiple be \(x\).
- Number of green stones = \(5x\)
- Number of black stones = \(8x\)
We are given that the actual number of black stones is 24. So, we can set up an equation:
\(8x = 24\)
Now, we need to solve for \(x\) to find the value of the common multiple:
\(x = \frac{24}{8}\)
\(x = 3\)
Now that we know the value of \(x\), we can find the actual number of green stones:
Number of green stones = \(5x = 5 \times 3 = 15\)
So, there are 15 green stones in the box.
The problem asks for the total number of stones. The total number of stones is the sum of the number of green stones and the number of black stones.
Total stones = Number of green stones + Number of black stones
Total stones = \(15 + 24\)
Total stones = \(39\)
Summary of Calculation
| Item | Ratio Part | Calculation | Number of Stones |
|---|---|---|---|
| Green Stones | 5 | \(5 \times 3\) | 15 |
| Black Stones | 8 | \(8 \times 3\) (Given 24, solved for \(x=3\)) | 24 |
| Total Stones | \(5+8=13\) | \(15 + 24\) | 39 |
Thus, the total number of stones in the box is 39.
Revision Table: Ratio of Stones
| Concept | Description | Application in Problem |
|---|---|---|
| Ratio | A comparison of two quantities. Written as \(a:b\) or \(\frac{a}{b}\). | Given as 5:8 for green to black stones. |
| Proportion | An equation stating that two ratios are equal. | Used to find the value of the ratio unit (\(x\)). \(\frac{\text{Green}}{\text{Black}} = \frac{5}{8}\). If Black is 24, Green is found via proportion. |
| Finding Unknowns | Using given information to find missing values in a ratio or proportion. | We used the number of black stones (24) and the ratio (8 part) to find the value of one ratio unit, then calculated green stones (5 parts). |
| Total Quantity | The sum of all individual quantities. | Calculated by adding the number of green stones and black stones. |
Additional Information: Understanding Ratios and Proportions
Ratios and proportions are fundamental concepts in mathematics used to compare quantities and solve problems involving scaling. A ratio like 5:8 tells us the relative size of two groups. It doesn't necessarily mean there are exactly 5 green and 8 black stones, but rather that the number of green stones divided by the number of black stones will always simplify to 5/8.
When we are given the actual number of one part of the ratio, we can use this information to find the actual numbers for all parts. In this problem, knowing that 24 black stones correspond to the '8' part of the ratio allowed us to figure out that each 'unit' of the ratio represents 3 stones (\(24 \div 8 = 3\)). Once we know this unit value (often called the constant of proportionality or represented by \(x\)), we can multiply it by the other parts of the ratio to find their actual quantities.
- If the ratio is \(a:b\), the quantities can be written as \(ax\) and \(bx\).
- If the actual value of \(bx\) is given, we can solve for \(x\).
- Then, we can find the actual value of \(ax\).
- The total quantity is \(ax + bx\).
This method is widely applicable in various problems involving scaling, mixing, sharing amounts in a given ratio, and converting units.
Important Questions from Ratio and Proportion
The cost of a diamond is directly proportional to the square of its weight. The cost of a 14 gm diamond is Rs. 2560. This diamond got broken down into two pieces in the ratio of 5 ∶ 9. How much loss percent is incurred due to this breakage ? (Correct to two decimal places)
Atul purchased Bread costing Rs.20 and gave a 100 rupee note to the shopkeeper. The shopkeeper gave the balance money in coins of denomination Rs.2, Rs.5 and Rs.10. If these coins are in the ratio 5 ∶ 4 ∶ 1, then how many Rs.5 coins did the shopkeeper give?
A person divides a certain amount among his three sons in the ratio of 3 ∶ 4 ∶ 5. If he had divided this amount in the ratio of 1/3,1/4,1/5, his son, who had got the lowest share earlier, would get Rs.1,188 more. Find the amount (in Rs).
In a school 3/8 of the number of students are girls and the rest are boys. One-third of the number of boys are below 10 years and 2/3 the number if girls are also below 10 years. If the number of students of age 10 or more years is 260. then the number of boys in the school is:
If a : b : c = \(\frac{1}{4} : \frac{1}{3} : \frac{1}{2}, \) then \( \ \frac{a}{b} : \frac{b}{c} : \frac{c}{a} = ?\)
