Solving the Ball Ratio Problem
We are given that the original ratio of red, blue, and green balls is 4:5:6. Let the original number of balls be represented as $4x$, $5x$, and $6x$ respectively, where $x$ is a common multiplier.
Adjusting Ball Quantities
The problem states that 6 red balls are removed and 3 green balls are added. The number of blue balls remains unchanged.
- Original Red Balls: $4x$
- New Red Balls: $4x - 6$
- Original Blue Balls: $5x$
- New Blue Balls: $5x$
- Original Green Balls: $6x$
- New Green Balls: $6x + 3$
New Ratio Calculation
The new ratio of red, blue, and green balls is given as 2:5:7. We can set up equations using the new quantities and the new ratio.
The ratio of new blue balls to new red balls is $5:2$. So, we can write:
$ \frac{5x}{4x - 6} = \frac{5}{2} $Now, we solve this equation for $x$. Cross-multiplying gives:
$ 2 \times (5x) = 5 \times (4x - 6) $ $ 10x = 20x - 30 $Rearranging the terms to solve for $x$:
$ 30 = 20x - 10x $ $ 30 = 10x $ $ x = \frac{30}{10} $ $ x = 3 $Verifying the Multiplier
Let's verify this value of $x$ using the ratio of new blue balls to new green balls, which is $5:7$.
New Blue Balls = $5x = 5 \times 3 = 15$
New Green Balls = $6x + 3 = 6 \times 3 + 3 = 18 + 3 = 21$
The ratio is $\frac{15}{21}$, which simplifies to $\frac{5}{7}$ by dividing both numerator and denominator by 3. This matches the given new ratio.
Finding Original Red Balls
The question asks for the original number of red balls. The original number of red balls was represented as $4x$. Using the value $x=3$:
$ \text{Original Red Balls} = 4x = 4 \times 3 = 12 $Therefore, there were originally 12 red balls in the bag.
Similar Questions
- A ribbon is cut into three pieces in the ratio 2:3:5. If the length of the longest piece is 20 cm, what is the total length of the ribbon?
- Three numbers are in the ratio 2:3:5. If their average is 100, what is the difference between the largest and the smallest number?
- Three numbers are such that the second is 150% of the first, and the third is 80% of the second. What is the ratio of the first to the third?
- Three friends share a winning prize in the ratio 2:3:4. If the total prize is Rs. 9000, how much more does the person with the highest share get than the lowest?
- A town has males and females in a ratio of 3:2. After 5 years, the male population increases by 10%, and the female population by 25%. What is the new ratio?
- The weights of three friends are in the ratio 7:8:9. If each gains 5 kg, the new ratio becomes 12:13:14. What is the original weight of the heaviest friend?
- Two numbers are in the ratio 6:5. If the difference between them is 11, find the numbers.
- A recipe requires 0.625 liters of milk for 5 servings. How many servings can be made with 3.75 liters of milk?
- If $x:y = 2:5$, then what is $(2x + 3y):(x + y)$?
Rs. 640 shared among 8 people. 2 back out. What's new share per person?
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 \( \ \frac{a}{b} : \frac{b}{c} : \frac{c}{a} = ?\)
