what happens when a frog double - parks on a lily pad? write the letter of each answer in the box containing the exercise number. solve the equation. 1. $x + 36 = 4x$ 2. $6a+12 = 2(3a - 8)$ 3. $\frac{3}{2}p-14 = p + 13$ 4. $7-4.9t = 15 + 7.6t$ 5. $\frac{1}{3}(12f - 3)=4f - 1$ 6. $\frac{1}{3}(b + 6)=\frac{1}{4}b + 8$ 7. $\frac{3}{5}(2m - 10)=\frac{2}{3}m + 10$ 8. $8.2(s + 4)=6.7s + 5.2$ 9. on monday, you run on a treadmill for $\frac{1}{2}$ hour at $x$ miles per hour. on tuesday, you walk the same distance on the treadmill, at 2 miles per hour slower, and it takes you $\frac{3}{4}$ hour. how many miles did you run on the treadmill on monday? 10. jess spent $7x$ minutes on the computer. her sister spent $(5x + 10)$ minutes on the computer, which was the same amount of time jess spent. how many minutes was jess on the computer? 11. a rectangle is 6 units wide and $(x - 8)$ units long. it has the same area as a triangle with a height of 7 units and a base of $(x - 3)$ units. what is the area of the rectangle (in square units)? answers: y. 72 a. - 18.4 t. 42 o. no solution a. 35 s. 54 a. 12 d. - 0.64 w. 3 i. infinitely many solutions t. 30

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Accepted Answer

1. **For the equation $x + 36=4x$**:
   - **Step 1: Move $x$ - terms to one side**
     - Subtract $x$ from both sides of the equation. $x+36 - x=4x - x$, which simplifies to $36 = 3x$.
   - **Step 2: Solve for $x$**
     - Divide both sides by 3. $\frac{36}{3}=\frac{3x}{3}$, so $x = 12$. The answer - letter is A.
2. **For the equation $6a + 12=2(3a - 8)$**:
   - **Step 1: Expand the right - hand side**
     - Using the distributive property $2(3a - 8)=6a-16$. So the equation becomes $6a + 12=6a - 16$.
   - **Step 2: Analyze the result**
     - Subtract $6a$ from both sides: $6a + 12-6a=6a - 16-6a$, we get $12=-16$, which is a contradiction. So the equation has no solution. The answer - letter is O.
3. **For the equation $\frac{3}{2}p-14=p + 13$**:
   - **Step 1: Move $p$ - terms to one side**
     - Subtract $p$ from both sides: $\frac{3}{2}p-14 - p=p + 13 - p$, which simplifies to $\frac{3}{2}p-p=13 + 14$. Since $\frac{3}{2}p-p=\frac{3p-2p}{2}=\frac{1}{2}p$, the equation is $\frac{1}{2}p=27$.
   - **Step 2: Solve for $p$**
     - Multiply both sides by 2: $p = 54$. The answer - letter is S.
4. **For the equation $7-4.9t=15 + 7.6t$**:
   - **Step 1: Move $t$ - terms to one side**
     - Add $4.9t$ to both sides: $7-4.9t + 4.9t=15 + 7.6t+4.9t$, which gives $7=15 + 12.5t$.
   - **Step 2: Isolate $t$**
     - Subtract 15 from both sides: $7-15=15 + 12.5t-15$, so $-8 = 12.5t$. Then divide both sides by 12.5: $t=\frac{-8}{12.5}=-0.64$. The answer - letter is D.
5. **For the equation $\frac{1}{3}(12f-3)=4f - 1$**:
   - **Step 1: Expand the left - hand side**
     - Using the distributive property, $\frac{1}{3}(12f-3)=4f - 1$. The left - hand side simplifies to $4f-1$, so $4f-1=4f - 1$. This is an identity, and there are infinitely many solutions. The answer - letter is I.
6. **For the equation $\frac{1}{3}(b + 6)=\frac{1}{4}b + 8$**:
   - **Step 1: Expand the left - hand side**
     - $\frac{1}{3}(b + 6)=\frac{1}{3}b+2$. So the equation is $\frac{1}{3}b + 2=\frac{1}{4}b + 8$.
   - **Step 2: Move $b$ - terms to one side**
     - Subtract $\frac{1}{4}b$ from both sides: $\frac{1}{3}b-\frac{1}{4}b+2=\frac{1}{4}b + 8-\frac{1}{4}b$. Since $\frac{1}{3}b-\frac{1}{4}b=\frac{4b - 3b}{12}=\frac{1}{12}b$, the equation becomes $\frac{1}{12}b+2 = 8$.
   - **Step 3: Isolate $b$**
     - Subtract 2 from both sides: $\frac{1}{12}b+2-2=8 - 2$, so $\frac{1}{12}b=6$. Multiply both sides by 12 to get $b = 72$. The answer - letter is Y.
7. **For the equation $\frac{3}{5}(2m-10)=\frac{2}{3}m + 10$**:
   - **Step 1: Expand the left - hand side**
     - $\frac{3}{5}(2m-10)=\frac{6}{5}m-6$. So the equation is $\frac{6}{5}m-6=\frac{2}{3}m + 10$.
   - **Step 2: Move $m$ - terms to one side**
     - Subtract $\frac{2}{3}m$ from both sides: $\frac{6}{5}m-\frac{2}{3}m-6=\frac{2}{3}m + 10-\frac{2}{3}m$. Find a common denominator for the $m$ - terms: $\frac{18m - 10m}{15}-6 = 10$, or $\frac{8}{15}m-6 = 10$.
   - **Step 3: Isolate $m$**
     - Add 6 to both sides: $\frac{8}{15}m-6 + 6=10 + 6$, so $\frac{8}{15}m=16$. Multiply both sides by $\frac{15}{8}$: $m = 30$. The answer - letter is T.
8. **For the equation $8.2(s + 4)=6.7s + 5.2$**:
   - **Step 1: Expand the left - hand side**
     - $8.2(s + 4)=8.2s+32.8$. So the equation is $8.2s+32.8=6.7s + 5.2$.
   - **Step 2: Move $s$ - terms to one side**
     - Subtract $6.7s$ from both sides: $8.2s-6.7s+32.8=6.7s + 5.2-6.7s$, which gives $1.5s+32.8 = 5.2$.
   - **Step 3: Isolate $s$**
     - Subtract 32.8 from both sides: $1.5s+32.8-32.8=5.2 - 32.8$, so $1.5s=-27.6$. Divide both sides by 1.5: $s=-18.4$. The answer - letter is A.
9. **For the treadmill problem**:
   - **Step 1: Set up the distance equation**
     - The distance formula is $d = rt$ (distance = rate×time). On Monday, the distance $d_1=\frac{1}{2}x$. On Tuesday, the rate is $(x - 2)$ and the time is $\frac{3}{4}$, so $d_2=\frac{3}{4}(x - 2)$. Since $d_1 = d_2$, we have the equation $\frac{1}{2}x=\frac{3}{4}(x - 2)$.
   - **Step 2: Expand the right - hand side**
     - $\frac{1}{2}x=\frac{3}{4}x-\frac{3}{2}$.
   - **Step 3: Move $x$ - terms to one side**
     - Subtract $\frac{3}{4}x$ from both sides: $\frac{1}{2}x-\frac{3}{4}x=\frac{3}{4}x-\frac{3}{2}-\frac{3}{4}x$. $\frac{2x-3x}{4}=-\frac{3}{2}$, or $-\frac{1}{4}x=-\frac{3}{2}$.
   - **Step 4: Solve for $x$**
     - Multiply both sides by - 4: $x = 6$. The distance on Monday is $d=\frac{1}{2}x$, so $d = 3$ miles. The answer - letter is W.
10. **For the computer - time problem**:
   - **Step 1: Set up the equation**
     - Since Jess and her sister spent the same amount of time on the computer, $7x=5x + 10$.
   - **Step 2: Move $x$ - terms to one side**
     - Subtract $5x$ from both sides: $7x-5x=5x + 10-5x$, so $2x=10$.
   - **Step 3: Solve for $x$**
     - Divide both sides by 2: $x = 5$. Jess spent $7x$ minutes, so $7\times5 = 35$ minutes. The answer - letter is A.
11. **For the rectangle - triangle area problem**:
   - **Step 1: Set up the area equation**
     - The area of the rectangle $A_{r}=6(x - 8)$, and the area of the triangle $A_{t}=\frac{1}{2}\times7\times(x - 3)$. Since $A_{r}=A_{t}$, we have $6(x - 8)=\frac{7}{2}(x - 3)$.
   - **Step 2: Expand both sides**
     - $6x-48=\frac{7}{2}x-\frac{21}{2}$.
   - **Step 3: Move $x$ - terms to one side**
     - Subtract $\frac{7}{2}x$ from both sides: $6x-\frac{7}{2}x-48=\frac{7}{2}x-\frac{21}{2}-\frac{7}{2}x$. $\frac{12x - 7x}{2}-48=-\frac{21}{2}$, or $\frac{5}{2}x-48=-\frac{21}{2}$.
   - **Step 4: Isolate $x$**
     - Add 48 to both sides: $\frac{5}{2}x-48 + 48=-\frac{21}{2}+48$. $\frac{5}{2}x=\frac{-21 + 96}{2}=\frac{75}{2}$. Multiply both sides by $\frac{2}{5}$: $x = 15$.
   - **Step 5: Find the area of the rectangle**
     - The area of the rectangle $A = 6(x - 8)$, substitute $x = 15$: $A=6\times(15 - 8)=42$. The answer - letter is T.

# Answer:
1. A. 12
2. O. no solution
3. S. 54
4. D. - 0.64
5. I. infinitely many solutions
6. Y. 72
7. T. 30
8. A. - 18.4
9. W. 3
10. A. 35
11. T. 42

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