The sum of three numbers is 67. If the ratio of the first number to the second number is 3:5 and that of the second to the third is 4:7, then what is the second number?
(a)20
(b)24
(c)18
(d)16

Options

A. 20
B. 24
C. 18
D. 16

Answer

To solve the problem step by step, we will use the information given about the ratios and the total sum of the three numbers.

### Step 1: Define the Variables
Let the three numbers be:
- First number = \( a \)
- Second number = \( b \)
- Third number = \( c \)

### Step 2: Set Up the Ratios
According to the problem:
- The ratio of the first number to the second number is \( 3:5 \). This can be expressed as:
\[ \frac{a}{b} = \frac{3}{5} \implies a = \frac{3}{5}b \]

- The ratio of the second number to the third number is \( 4:7 \). This can be expressed as:
\[ \frac{b}{c} = \frac{4}{7} \implies c = \frac{7}{4}b \]

### Step 3: Express All Numbers in Terms of \( b \)
Now we can express \( a \) and \( c \) in terms of \( b \):
- From \( a = \frac{3}{5}b \)
- From \( c = \frac{7}{4}b \)

### Step 4: Write the Sum of the Numbers
According to the problem, the sum of the three numbers is 67:
\[ a + b + c = 67 \]
Substituting the expressions for \( a \) and \( c \):
\[ \frac{3}{5}b + b + \frac{7}{4}b = 67 \]

### Step 5: Find a Common Denominator
To combine the terms, we need a common denominator. The least common multiple of 5 and 4 is 20. Thus, we convert each term:
- \( \frac{3}{5}b = \frac{12}{20}b \)
- \( b = \frac{20}{20}b \)
- \( \frac{7}{4}b = \frac{35}{20}b \)

Now we can rewrite the equation:
\[ \frac{12}{20}b + \frac{20}{20}b + \frac{35}{20}b = 67 \]

### Step 6: Combine the Terms
Combine the fractions:
\[ \frac{12 + 20 + 35}{20}b = 67 \]
\[ \frac{67}{20}b = 67 \]

### Step 7: Solve for \( b \)
To isolate \( b \), multiply both sides by 20:
\[ 67b = 67 \times 20 \]
\[ 67b = 1340 \]
Now divide both sides by 67:
\[ b = \frac{1340}{67} = 20 \]

### Conclusion
The second number \( b \) is 20.

### Answer
The second number is \( \boxed{20} \).

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