In a bag containing red, green and blue pens, the ratio of red, blue and green pens, in the given order, was 7 ∶ 4 ∶ 9. If the total number of pens in the bag was 320, how many of them were red?
112
Understanding Ratio and Pen Calculation
This question involves calculating the number of items (pens) belonging to a specific category (red) when the total number of items and the ratio of different categories are given. The key is to understand how ratios represent parts of a whole and how to use the total number to find the value of each part in the ratio.
Breaking Down the Ratio of Pens
We are given the ratio of red, blue, and green pens as 7 ∶ 4 ∶ 9. This means that for every 7 red pens, there are 4 blue pens and 9 green pens. These numbers represent proportional parts of the total collection of pens.
- Red pens correspond to a ratio of 7 parts.
- Blue pens correspond to a ratio of 4 parts.
- Green pens correspond to a ratio of 9 parts.
Calculating the Total Number of Ratio Parts
To find out how many total parts the pens are divided into according to the ratio, we sum the individual ratio parts:
\(\text{Total ratio parts} = \text{Ratio of red} + \text{Ratio of blue} + \text{Ratio of green}\)
\(\text{Total ratio parts} = 7 + 4 + 9\)
\(\text{Total ratio parts} = 20\)
So, the total collection of 320 pens is divided into 20 equal ratio parts.
Determining the Value of One Ratio Part
We know the total number of pens is 320, and this total corresponds to 20 ratio parts. To find the number of pens that makes up one ratio part, we divide the total number of pens by the total number of ratio parts:
\(\text{Value of one ratio part} = \frac{\text{Total number of pens}}{\text{Total ratio parts}}\)
\(\text{Value of one ratio part} = \frac{320}{20}\)
\(\text{Value of one ratio part} = 16\)
This means each part in the ratio 7 ∶ 4 ∶ 9 represents 16 pens.
Calculating the Number of Red Pens
The ratio of red pens is given as 7. Since each ratio part is equal to 16 pens, the number of red pens is found by multiplying the red pen ratio part by the value of one ratio part:
\(\text{Number of red pens} = \text{Ratio of red} \times \text{Value of one ratio part}\)
\(\text{Number of red pens} = 7 \times 16\)
\(\text{Number of red pens} = 112\)
Therefore, there were 112 red pens in the bag.
| Pen Color | Ratio Part | Calculation (Ratio × Value of one part) | Number of Pens |
|---|---|---|---|
| Red | 7 | \(7 \times 16\) | 112 |
| Blue | 4 | \(4 \times 16\) | 64 |
| Green | 9 | \(9 \times 16\) | 144 |
| Total | 20 | \(112 + 64 + 144 = 320\) |
The calculated numbers for each color sum up to the total number of pens, 320, which confirms our calculation is correct.
Revision Table: Pen Ratio and Total Pens
| Concept | Explanation | Formula/Calculation |
|---|---|---|
| Ratio | Compares the relative sizes of two or more values. | Given as a:b:c |
| Total Ratio Parts | Sum of all individual ratio parts. | Sum = a + b + c |
| Value of One Ratio Part | Total quantity divided by the total ratio parts. | Value = \(\frac{\text{Total Quantity}}{\text{Total Ratio Parts}}\) |
| Quantity of a Specific Item | Ratio part of the specific item multiplied by the value of one ratio part. | Quantity = Ratio part × Value of one part |
Additional Information: Working with Ratios
Ratios are used in many real-world situations to show how quantities are related proportionally. When dealing with ratios and a total quantity, you can always find the amount for each part by following the steps outlined above. This method is applicable whether you are mixing ingredients, sharing money, or calculating proportions of items in a collection.
Key points to remember:
- Always sum the ratio parts to find the total parts.
- Divide the total quantity by the sum of ratio parts to find the value of one part.
- Multiply the value of one part by the ratio part of the item you are interested in.
- Ensure the sum of the calculated quantities for each item equals the total quantity given in the problem.
Understanding ratios is fundamental for solving many types of quantitative problems.
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Important Questions from Simple Ratios
The ratio of two numbers is 9 : 5. If 8 is added to the larger number and 4 is subtracted from the smaller number, the greater number becomes twice the smaller number. The larger number is:
Divide 500 into two parts such that the ratio of one to the other are in 5 : 3?
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The monthly salaries of an officer and a clerk are in the ratio 11 : 4. If the monthly salary of the officer increases by ₹7,000 and that of the clerk by ₹3,000, then the ratio becomes 19 : 7. What was the initial salary (in ₹) of the officer?
If X : Y = 7 : 5 and Y : Z = 7 : 11, then what is the ratio of X : Y : Z?
