Translating Algebraic Phrases (Complex Version) (A) - Free Printable

The task involves translating verbal phrases into algebraic expressions. Below, I will explain the solution for each problem step by step.

---

1. The product of a number \( n \) and six is divided by thirty-one.

- Translation: The product of \( n \) and 6 is \( 6n \).
- Dividing this by 31 gives: \( \frac{6n}{31} \).
- Answer: \( \frac{6n}{31} \).

---

2. The product of a number \( b \) and itself.

- Translation: The product of \( b \) and itself is \( b \times b \), which is \( b^2 \).
- Answer: \( b^2 \).

---

3. The square root of the product of a number \( z \) and itself.

- Translation: The product of \( z \) and itself is \( z \times z = z^2 \).
- The square root of \( z^2 \) is \( \sqrt{z^2} \), which simplifies to \( z \) (assuming \( z \geq 0 \)).
- Answer: \( z \).

---

4. The sum of a number \( p \) and its cube.

- Translation: The cube of \( p \) is \( p^3 \).
- The sum of \( p \) and \( p^3 \) is \( p + p^3 \).
- Answer: \( p + p^3 \).

---

5. The square of the quotient of a number \( v \) and five.

- Translation: The quotient of \( v \) and 5 is \( \frac{v}{5} \).
- The square of this quotient is \( \left( \frac{v}{5} \right)^2 \).
- Answer: \( \left( \frac{v}{5} \right)^2 \).

---

6. The sum of a number \( q \) and sixty-four divided by eighty-three.

- Translation: The sum of \( q \) and 64 is \( q + 64 \).
- Dividing this sum by 83 gives: \( \frac{q + 64}{83} \).
- Answer: \( \frac{q + 64}{83} \).

---

7. The difference of the square root of a number \( r \) and thirty-seven.

- Translation: The square root of \( r \) is \( \sqrt{r} \).
- The difference between \( \sqrt{r} \) and 37 is \( \sqrt{r} - 37 \).
- Answer: \( \sqrt{r} - 37 \).

---

8. The difference of a number \( m \) and itself.

- Translation: The difference between \( m \) and \( m \) is \( m - m \), which equals 0.
- Answer: \( 0 \).

---

9. A number \( y \) squared plus twice the same number minus seventy-four.

- Translation: Squaring \( y \) gives \( y^2 \).
- Twice the number \( y \) is \( 2y \).
- Adding these together and subtracting 74 gives: \( y^2 + 2y - 74 \).
- Answer: \( y^2 + 2y - 74 \).

---

10. The square root of the difference of a number \( d \) and seventy-nine.

- Translation: The difference of \( d \) and 79 is \( d - 79 \).
- The square root of this difference is \( \sqrt{d - 79} \).
- Answer: \( \sqrt{d - 79} \).

---

11. The inverse of a number \( h \).

- Translation: The inverse of \( h \) is \( \frac{1}{h} \).
- Answer: \( \frac{1}{h} \).

---

12. Two times the cube of the difference of a number \( w \) and forty-two.

- Translation: The difference of \( w \) and 42 is \( w - 42 \).
- Cubing this difference gives \( (w - 42)^3 \).
- Multiplying by 2 gives: \( 2(w - 42)^3 \).
- Answer: \( 2(w - 42)^3 \).

---

13. The sum of a number \( f \) and itself.

- Translation: The sum of \( f \) and \( f \) is \( f + f \), which simplifies to \( 2f \).
- Answer: \( 2f \).

---

14. Four times the square of a number \( t \) divided by fourteen more than \( e \).

- Translation: The square of \( t \) is \( t^2 \).
- Four times this square is \( 4t^2 \).
- Fourteen more than \( e \) is \( e + 14 \).
- Dividing \( 4t^2 \) by \( e + 14 \) gives: \( \frac{4t^2}{e + 14} \).
- Answer: \( \frac{4t^2}{e + 14} \).

---

15. The quotient of a number \( g \) and itself.

- Translation: The quotient of \( g \) and \( g \) is \( \frac{g}{g} \), which simplifies to 1 (assuming \( g \neq 0 \)).
- Answer: \( 1 \).

---

16. Fifteen times the sum of a number \( x \) and twenty-nine.

- Translation: The sum of \( x \) and 29 is \( x + 29 \).
- Fifteen times this sum is \( 15(x + 29) \).
- Answer: \( 15(x + 29) \).

---

17. The sum of one ninth of a number \( x \) and fifty-one.

- Translation: One ninth of \( x \) is \( \frac{x}{9} \).
- Adding 51 to this gives: \( \frac{x}{9} + 51 \).
- Answer: \( \frac{x}{9} + 51 \).

---

18. The sum of a number \( k \) and eighty-three to the power of four.

- Translation: The sum of \( k \) and 83 is \( k + 83 \).
- Raising this sum to the power of 4 gives: \( (k + 83)^4 \).
- Answer: \( (k + 83)^4 \).

---

19. A number \( j \) multiplied by itself thirty-four times.

- Translation: Multiplying \( j \) by itself 34 times means raising \( j \) to the power of 34, which is \( j^{34} \).
- Answer: \( j^{34} \).

---

20. One half of a number \( c \) is subtracted from nineteen.

- Translation: One half of \( c \) is \( \frac{c}{2} \).
- Subtracting this from 19 gives: \( 19 - \frac{c}{2} \).
- Answer: \( 19 - \frac{c}{2} \).

---

Final Answer:

\[
\boxed{
\begin{aligned}
1. & \ \frac{6n}{31} \\
2. & \ b^2 \\
3. & \ z \\
4. & \ p + p^3 \\
5. & \ \left( \frac{v}{5} \right)^2 \\
6. & \ \frac{q + 64}{83} \\
7. & \ \sqrt{r} - 37 \\
8. & \ 0 \\
9. & \ y^2 + 2y - 74 \\
10. & \ \sqrt{d - 79} \\
11. & \ \frac{1}{h} \\
12. & \ 2(w - 42)^3 \\
13. & \ 2f \\
14. & \ \frac{4t^2}{e + 14} \\
15. & \ 1 \\
16. & \ 15(x + 29) \\
17. & \ \frac{x}{9} + 51 \\
18. & \ (k + 83)^4 \\
19. & \ j^{34} \\
20. & \ 19 - \frac{c}{2}
\end{aligned}
}
\]