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Sagot :
Certainly! Let's go through each part of the problem step-by-step to find the solutions.
### Part 1: Sumitra's Coins
Sumitra has a total of Rs. 34 in 50-paise and 25-paise coins. The number of 25-paise coins is twice the number of 50-paise coins. We need to find out how many coins of each kind she has.
1. Let [tex]\( x \)[/tex] be the number of 50-paise coins.
2. Since the number of 25-paise coins is twice the number of 50-paise coins, we can express the number of 25-paise coins as [tex]\( 2x \)[/tex].
3. The total amount in Rs. needs to be converted into paise for easier calculation.
[tex]\[ \text{Total amount} = 34 \text{ Rs} = 3400 \text{ paise} \][/tex]
4. Now, we need to express the total value of 50-paise and 25-paise coins in terms of paise:
[tex]\[ \text{Value from 50-paise coins} = 50 \text{ paise} \times x \][/tex]
[tex]\[ \text{Value from 25-paise coins} = 25 \text{ paise} \times 2x \][/tex]
5. Combining these to set up an equation for the total amount of paise:
[tex]\[ 50x + 25 \times 2x = 3400 \][/tex]
[tex]\[ 50x + 50x = 3400 \][/tex]
[tex]\[ 100x = 3400 \][/tex]
[tex]\[ x = 34 \][/tex]
6. Therefore, the number of 50-paise coins is [tex]\( x = 34 \)[/tex].
7. The number of 25-paise coins is [tex]\( 2x = 2 \times 34 = 68 \)[/tex].
So, Sumitra has 34 coins of 50-paise and 68 coins of 25-paise.
### Part 2: Raju's Age Problem
Raju is 19 years younger than his cousin. After 5 years, their ages will be in the ratio 2:3. We need to find their current ages.
1. Let [tex]\( r \)[/tex] be Raju's current age.
2. Therefore, Raju's cousin's current age, being 19 years older, is [tex]\( r + 19 \)[/tex].
3. After 5 years, their ages will be:
[tex]\[ \text{Raju's age after 5 years} = r + 5 \][/tex]
[tex]\[ \text{Cousin's age after 5 years} = (r + 19) + 5 = r + 24 \][/tex]
4. According to the problem, the ratio of their ages after 5 years will be 2:3:
[tex]\[ \frac{r + 5}{r + 24} = \frac{2}{3} \][/tex]
5. We can solve this relationship by cross-multiplying:
[tex]\[ 3(r + 5) = 2(r + 24) \][/tex]
[tex]\[ 3r + 15 = 2r + 48 \][/tex]
[tex]\[ 3r - 2r = 48 - 15 \][/tex]
[tex]\[ r = 33 \][/tex]
6. Therefore, Raju’s current age is [tex]\( r = 33 \)[/tex].
7. His cousin’s current age is [tex]\( r + 19 \)[/tex], which simplifies to:
[tex]\[ 33 + 19 = 52 \][/tex]
So, Raju is currently 33 years old, and his cousin is 52 years old.
### Summary:
- Sumitra has 34 coins of 50-paise and 68 coins of 25-paise.
- Raju is 33 years old, and his cousin is 52 years old.
### Part 1: Sumitra's Coins
Sumitra has a total of Rs. 34 in 50-paise and 25-paise coins. The number of 25-paise coins is twice the number of 50-paise coins. We need to find out how many coins of each kind she has.
1. Let [tex]\( x \)[/tex] be the number of 50-paise coins.
2. Since the number of 25-paise coins is twice the number of 50-paise coins, we can express the number of 25-paise coins as [tex]\( 2x \)[/tex].
3. The total amount in Rs. needs to be converted into paise for easier calculation.
[tex]\[ \text{Total amount} = 34 \text{ Rs} = 3400 \text{ paise} \][/tex]
4. Now, we need to express the total value of 50-paise and 25-paise coins in terms of paise:
[tex]\[ \text{Value from 50-paise coins} = 50 \text{ paise} \times x \][/tex]
[tex]\[ \text{Value from 25-paise coins} = 25 \text{ paise} \times 2x \][/tex]
5. Combining these to set up an equation for the total amount of paise:
[tex]\[ 50x + 25 \times 2x = 3400 \][/tex]
[tex]\[ 50x + 50x = 3400 \][/tex]
[tex]\[ 100x = 3400 \][/tex]
[tex]\[ x = 34 \][/tex]
6. Therefore, the number of 50-paise coins is [tex]\( x = 34 \)[/tex].
7. The number of 25-paise coins is [tex]\( 2x = 2 \times 34 = 68 \)[/tex].
So, Sumitra has 34 coins of 50-paise and 68 coins of 25-paise.
### Part 2: Raju's Age Problem
Raju is 19 years younger than his cousin. After 5 years, their ages will be in the ratio 2:3. We need to find their current ages.
1. Let [tex]\( r \)[/tex] be Raju's current age.
2. Therefore, Raju's cousin's current age, being 19 years older, is [tex]\( r + 19 \)[/tex].
3. After 5 years, their ages will be:
[tex]\[ \text{Raju's age after 5 years} = r + 5 \][/tex]
[tex]\[ \text{Cousin's age after 5 years} = (r + 19) + 5 = r + 24 \][/tex]
4. According to the problem, the ratio of their ages after 5 years will be 2:3:
[tex]\[ \frac{r + 5}{r + 24} = \frac{2}{3} \][/tex]
5. We can solve this relationship by cross-multiplying:
[tex]\[ 3(r + 5) = 2(r + 24) \][/tex]
[tex]\[ 3r + 15 = 2r + 48 \][/tex]
[tex]\[ 3r - 2r = 48 - 15 \][/tex]
[tex]\[ r = 33 \][/tex]
6. Therefore, Raju’s current age is [tex]\( r = 33 \)[/tex].
7. His cousin’s current age is [tex]\( r + 19 \)[/tex], which simplifies to:
[tex]\[ 33 + 19 = 52 \][/tex]
So, Raju is currently 33 years old, and his cousin is 52 years old.
### Summary:
- Sumitra has 34 coins of 50-paise and 68 coins of 25-paise.
- Raju is 33 years old, and his cousin is 52 years old.
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