The sum of three numbers is 136. The ratio between the first and second numbers is 2:3, and between the second and third numbers is 5:3. The first number is:

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Accepted Answer

1. **Define variables based on ratios:**  
   - Let the first number be $$2y$$ and the second number be $$3y$$ (ratio 2:3).  
   - Let the second number be $$5z$$ and the third number be $$3z$$ (ratio 5:3).

2. **Equate the second number’s expressions:**  
   - Since $$3y = 5z$$, solve for $$y$$:  
     $$
     y = \frac{5z}{3}.
     $$

3. **Express all numbers in terms of $$z$$:**  
   - First number: $$2y = \frac{10z}{3}$$  
   - Second number: $$5z$$  
   - Third number: $$3z$$

4. **Set up the sum equation and solve for $$z$$:**  
   - Total sum:  
     $$
     \frac{10z}{3} + 5z + 3z = 136.
     $$  
   - Combine terms:  
     $$
     \frac{34z}{3} = 136 \implies z = \frac{136 	imes 3}{34} = 12.
     $$

5. **Calculate the first number:**  
   - Substitute $$z = 12$$ into $$2y = \frac{10z}{3}$$:  
     $$
     2y = \frac{10 	imes 12}{3} = 40.
     $$

Thus, the first number is **40**.

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