A and B have 460 coins altogether. If ` (3)/(4)` of A's number of coins is equal to ` ( 2)/(5)` of B's number of coins, find how many coins must B give to A so that they both have equal number of coins.

Answer

To solve the problem step-by-step, we will follow the information given and use algebra to find the number of coins A and B have, and then determine how many coins B must give to A for them to have an equal number of coins.

### Step 1: Set up the equations
Let A be the number of coins A has, and B be the number of coins B has. According to the problem, we have:
1. \( A + B = 460 \) (Equation 1)
2. \( \frac{3}{4}A = \frac{2}{5}B \) (Equation 2)

### Step 2: Rearranging Equation 2
From Equation 2, we can express A in terms of B:
\[ \frac{3}{4}A = \frac{2}{5}B \]
Multiplying both sides by 20 (the least common multiple of 4 and 5) to eliminate the fractions:
\[ 20 \cdot \frac{3}{4}A = 20 \cdot \frac{2}{5}B \]
This simplifies to:
\[ 15A = 8B \]
Now, we can express A in terms of B:
\[ A = \frac{8}{15}B \quad \text{(Equation 3)} \]

### Step 3: Substitute A in Equation 1
Now, substitute Equation 3 into Equation 1:
\[ \frac{8}{15}B + B = 460 \]
To combine the terms, convert B to a fraction:
\[ \frac{8}{15}B + \frac{15}{15}B = 460 \]
This simplifies to:
\[ \frac{23}{15}B = 460 \]

### Step 4: Solve for B
To find B, multiply both sides by the reciprocal of \(\frac{23}{15}\):
\[ B = 460 \cdot \frac{15}{23} \]
Calculating this gives:
\[ B = 300 \]

### Step 5: Find A
Now that we have B, we can find A using Equation 1:
\[ A + 300 = 460 \]
Thus:
\[ A = 460 - 300 = 160 \]

### Step 6: Determine how many coins B must give to A
Now, we need to find how many coins B must give to A so that they both have an equal number of coins. Let \( x \) be the number of coins B gives to A. After the transaction:
- A will have \( 160 + x \)
- B will have \( 300 - x \)

We want these two amounts to be equal:
\[ 160 + x = 300 - x \]
Solving for \( x \):
\[ 160 + x + x = 300 \]
\[ 2x = 300 - 160 \]
\[ 2x = 140 \]
\[ x = 70 \]

### Final Answer
B must give 70 coins to A.

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