1. **State the problem:** Raj has a total of 19 units of money composed of $1 coins, $2 notes, and $5 notes. He has 9 coins and notes in total. He has less than 4 coins. We need to find the total value of the $2 notes. 2. **Define variables:** Let $x$ = number of $1 coins$ Let $y$ = number of $2 notes$ Let $z$ = number of $5 notes$ 3. **Write equations from the problem:** Total number of coins and notes: $$x + y + z = 9$$ Total value: $$1 \cdot x + 2 \cdot y + 5 \cdot z = 19$$ Constraint on coins: $$x < 4$$ 4. **Solve the system:** From the first equation: $$x = 9 - y - z$$ Substitute into the value equation: $$1 \cdot (9 - y - z) + 2y + 5z = 19$$ Simplify: $$9 - y - z + 2y + 5z = 19$$ $$9 + y + 4z = 19$$ $$y + 4z = 10$$ 5. **Find integer solutions with $x < 4$:** From $x = 9 - y - z$ and $x < 4$: $$9 - y - z < 4$$ $$- y - z < -5$$ $$y + z > 5$$ 6. **Use $y + 4z = 10$ and $y + z > 5$ to find $y$ and $z$:** Try integer values for $z$: - If $z=1$, then $y + 4(1) = 10 \Rightarrow y = 6$, check $y + z = 6 + 1 = 7 > 5$ valid. - If $z=2$, then $y + 8 = 10 \Rightarrow y = 2$, check $y + z = 2 + 2 = 4 \not> 5$ invalid. - If $z=0$, then $y = 10$, $y + z = 10 > 5$ but then $x = 9 - 10 - 0 = -1$ invalid. So the only valid solution is $z=1$, $y=6$. 7. **Find $x$:** $$x = 9 - y - z = 9 - 6 - 1 = 2$$ Check $x < 4$ is true. 8. **Find total value of $2 notes:** $$2 \times y = 2 \times 6 = 12$$ **Final answer:** The total value of $2 notes is 12.