Question
Tahmar knows the formula for simple interest is I = Prt, where I represents the simple interest on an amount of money, P, for t years at r rate. She transforms the equation to isolate P : P = P : P equals StartFraction I Over r t EndFraction.. Using this formula, what is the amount of money, P, that will generate $20 at a 5% interest rate over 5 years?
Answers
GPT-4o mini
To find the amount of money \( P \) that will generate $20 at a 5% interest rate over 5 years, we can use the formula for simple interest, which is \( I = Prt \).
We want to isolate \( P \), so we rearrange the formula:
\[
P = \frac{I}{rt}
\]
Here, \( I \) represents the simple interest ($20), \( r \) is the interest rate (5% or 0.05 in decimal), and \( t \) is the time period in years (5 years).
Now we can plug in the values:
- \( I = 20 \)
- \( r = 0.05 \)
- \( t = 5 \)
Now substituting these values into the formula gives:
\[
P = \frac{20}{0.05 \times 5}
\]
Calculating the denominator:
\[
0.05 \times 5 = 0.25
\]
Now substituting back into the equation:
\[
P = \frac{20}{0.25}
\]
Calculating that gives:
\[
P = 80
\]
Thus, the amount of money \( P \) that will generate $20 at a 5% interest rate over 5 years is **$80**.
We want to isolate \( P \), so we rearrange the formula:
\[
P = \frac{I}{rt}
\]
Here, \( I \) represents the simple interest ($20), \( r \) is the interest rate (5% or 0.05 in decimal), and \( t \) is the time period in years (5 years).
Now we can plug in the values:
- \( I = 20 \)
- \( r = 0.05 \)
- \( t = 5 \)
Now substituting these values into the formula gives:
\[
P = \frac{20}{0.05 \times 5}
\]
Calculating the denominator:
\[
0.05 \times 5 = 0.25
\]
Now substituting back into the equation:
\[
P = \frac{20}{0.25}
\]
Calculating that gives:
\[
P = 80
\]
Thus, the amount of money \( P \) that will generate $20 at a 5% interest rate over 5 years is **$80**.
The tax rate as a percent, r, charged on an item can be determined using the formula StartFraction c Over p EndFraction minus 1 equals r. – 1 = r, where c is the final cost of the item and p is the price of the item before tax. Louise rewrites the equation to solve for the final cost of the item: c = p(1 + ). What is the final cost of a $40 item after an 8% tax is applied?
GPT-4o mini
To find the final cost of a $40 item after an 8% tax is applied, we can use the formula you provided. First, let's clarify the formula for the final cost \( c \):
\[
c = p(1 + r)
\]
where:
- \( p \) is the price of the item before tax,
- \( r \) is the tax rate expressed as a decimal.
Given:
- \( p = 40 \)
- \( r = 8\% = 0.08 \)
Now, substituting these values into the formula:
\[
c = 40(1 + 0.08)
\]
Calculating the expression in parentheses:
\[
1 + 0.08 = 1.08
\]
Now, substitute this back into the equation:
\[
c = 40 \times 1.08
\]
Calculating that gives:
\[
c = 43.20
\]
Therefore, the final cost of a $40 item after an 8% tax is applied is **$43.20**.
\[
c = p(1 + r)
\]
where:
- \( p \) is the price of the item before tax,
- \( r \) is the tax rate expressed as a decimal.
Given:
- \( p = 40 \)
- \( r = 8\% = 0.08 \)
Now, substituting these values into the formula:
\[
c = 40(1 + 0.08)
\]
Calculating the expression in parentheses:
\[
1 + 0.08 = 1.08
\]
Now, substitute this back into the equation:
\[
c = 40 \times 1.08
\]
Calculating that gives:
\[
c = 43.20
\]
Therefore, the final cost of a $40 item after an 8% tax is applied is **$43.20**.
The equation f = v + at represents the final velocity of an object, f, with an initial velocity, v, and an acceleration rate, a, over time, t. Which is an equivalent equation solved for a?
StartFraction f minus v Over t EndFraction equals a. = a
StartFraction f Over t EndFraction plus v equals a.– v = a
StartFraction f plus v Over t EndFraction equals a. = a
StartFraction f Over t EndFraction plus v equals a.+ v = a
StartFraction f minus v Over t EndFraction equals a. = a
StartFraction f Over t EndFraction plus v equals a.– v = a
StartFraction f plus v Over t EndFraction equals a. = a
StartFraction f Over t EndFraction plus v equals a.+ v = a
GPT-4o mini
To isolate \( a \) in the equation \( f = v + at \), we can follow these steps:
1. **Subtract \( v \) from both sides** to move the initial velocity term to the left side:
\[
f - v = at
\]
2. **Next, divide both sides by \( t \)** to solve for \( a\):
\[
a = \frac{f - v}{t}
\]
So, the equivalent equation solved for \( a \) is:
\[
a = \frac{f - v}{t}
\]
This matches the first option you provided:
\[
\frac{f - v}{t} = a
\]
Thus, the correct answer is:
**StartFraction f minus v Over t EndFraction equals a. = a**.
1. **Subtract \( v \) from both sides** to move the initial velocity term to the left side:
\[
f - v = at
\]
2. **Next, divide both sides by \( t \)** to solve for \( a\):
\[
a = \frac{f - v}{t}
\]
So, the equivalent equation solved for \( a \) is:
\[
a = \frac{f - v}{t}
\]
This matches the first option you provided:
\[
\frac{f - v}{t} = a
\]
Thus, the correct answer is:
**StartFraction f minus v Over t EndFraction equals a. = a**.
The equation a = StartFraction one-half EndFraction left-parenthesis b 1 plus b 1 right-parenthesis.(b1 + b2)h can be used to determine the area, a, of a trapezoid with height, h, and base lengths, b1 and b2. Which are equivalent equations? Check all that apply.
StartFraction 2 a Over h EndFraction minus b 2 equals b 1.– b2 = b1
StartFraction a Over 2 h EndFraction minus b 2 equals b 1. – b2 = b1
StartFraction 2 a minus b 2 Over h EndFraction equals b. = b1
StartFraction 2 a Over b 1 plus b 2 EndFraction equals h. = h
StartFraction a Over 2 left-parenthesis b 1 plus b 1 right-parenthesis EndFraction equals h.= h
StartFraction 2 a Over h EndFraction minus b 2 equals b 1.– b2 = b1
StartFraction a Over 2 h EndFraction minus b 2 equals b 1. – b2 = b1
StartFraction 2 a minus b 2 Over h EndFraction equals b. = b1
StartFraction 2 a Over b 1 plus b 2 EndFraction equals h. = h
StartFraction a Over 2 left-parenthesis b 1 plus b 1 right-parenthesis EndFraction equals h.= h
GPT-4o mini
To determine which of the equations provided are equivalent to the original equation \( a = \frac{1}{2}(b_1 + b_2)h \), we can manipulate the original equation and see how these options relate to it.
Starting with the formula:
\[
a = \frac{1}{2}(b_1 + b_2)h
\]
### Rearranging for \( b_1 \):
We can isolate \( b_1 \):
1. Multiply both sides by \( 2 \):
\[
2a = (b_1 + b_2)h
\]
2. Now, divide both sides by \( h \):
\[
\frac{2a}{h} = b_1 + b_2
\]
3. Finally, isolate \( b_1 \):
\[
b_1 = \frac{2a}{h} - b_2
\]
This means the equation \( \frac{2a}{h} - b_2 = b_1 \) is **equivalent**.
### Rearranging for \( h \):
Let's also rearrange for \( h \):
Starting from:
\[
2a = (b_1 + b_2)h
\]
4. Divide both sides by \( (b_1 + b_2) \):
\[
h = \frac{2a}{b_1 + b_2}
\]
This shows that the equation \( \frac{2a}{b_1 + b_2} = h \) is also **equivalent**.
### Evaluating the Options:
Now let's check each of the provided options:
1. **\( \frac{2a}{h} - b_2 = b_1 \)**: This is equivalent. **✔**
2. **\( \frac{a}{2h} - b_2 = b_1 \)**: This is not equivalent (does not match the steps). **✘**
3. **\( \frac{2a - b_2}{h} = b_1 \)**: This is also not equivalent. **✘**
4. **\( \frac{2a}{b_1 + b_2} = h \)**: This is equivalent. **✔**
5. **\( \frac{a}{2(b_1 + b_1)} = h \)**: This is not equivalent (there's a typo in the equation as \( b_1 + b_1 \) should be \( b_1 + b_2 \)). **✘**
### Summary of Equivalent Equations:
- **\( \frac{2a}{h} - b_2 = b_1 \)** (Equivalent)
- **\( \frac{2a}{b_1 + b_2} = h \)** (Equivalent)
The equivalent equations are options 1 and 4.
Starting with the formula:
\[
a = \frac{1}{2}(b_1 + b_2)h
\]
### Rearranging for \( b_1 \):
We can isolate \( b_1 \):
1. Multiply both sides by \( 2 \):
\[
2a = (b_1 + b_2)h
\]
2. Now, divide both sides by \( h \):
\[
\frac{2a}{h} = b_1 + b_2
\]
3. Finally, isolate \( b_1 \):
\[
b_1 = \frac{2a}{h} - b_2
\]
This means the equation \( \frac{2a}{h} - b_2 = b_1 \) is **equivalent**.
### Rearranging for \( h \):
Let's also rearrange for \( h \):
Starting from:
\[
2a = (b_1 + b_2)h
\]
4. Divide both sides by \( (b_1 + b_2) \):
\[
h = \frac{2a}{b_1 + b_2}
\]
This shows that the equation \( \frac{2a}{b_1 + b_2} = h \) is also **equivalent**.
### Evaluating the Options:
Now let's check each of the provided options:
1. **\( \frac{2a}{h} - b_2 = b_1 \)**: This is equivalent. **✔**
2. **\( \frac{a}{2h} - b_2 = b_1 \)**: This is not equivalent (does not match the steps). **✘**
3. **\( \frac{2a - b_2}{h} = b_1 \)**: This is also not equivalent. **✘**
4. **\( \frac{2a}{b_1 + b_2} = h \)**: This is equivalent. **✔**
5. **\( \frac{a}{2(b_1 + b_1)} = h \)**: This is not equivalent (there's a typo in the equation as \( b_1 + b_1 \) should be \( b_1 + b_2 \)). **✘**
### Summary of Equivalent Equations:
- **\( \frac{2a}{h} - b_2 = b_1 \)** (Equivalent)
- **\( \frac{2a}{b_1 + b_2} = h \)** (Equivalent)
The equivalent equations are options 1 and 4.
Nathaniel writes the general form of the equation gm = cm + rg for when the equation is solved for m. He uses the general form to solve the equation –3m = 4m – 15 for m. Which expression shows what Nathaniel will actually evaluate?
4 + 15 – 3
4 – 15 + 3
StartFraction negative 15 Over negative 3 minus 4 EndFraction.
–15 – negative 15 minus StartFraction negative 3 Over 4 EndFraction.
4 + 15 – 3
4 – 15 + 3
StartFraction negative 15 Over negative 3 minus 4 EndFraction.
–15 – negative 15 minus StartFraction negative 3 Over 4 EndFraction.
GPT-4o mini
To solve the equation \(-3m = 4m - 15\) for \(m\), let's first rearrange it to isolate \(m\).
1. Start with the original equation:
\[
-3m = 4m - 15
\]
2. Add \(3m\) to both sides to move all terms involving \(m\) to one side:
\[
0 = 4m + 3m - 15 \quad \Rightarrow \quad 0 = 7m - 15
\]
3. Next, add \(15\) to both sides to isolate the \(m\) term:
\[
15 = 7m
\]
4. Finally, divide both sides by \(7\) to solve for \(m\):
\[
m = \frac{15}{7}
\]
Now, let's express this process using the expressions provided to evaluate.
From our rearrangement, we reached \(m = \frac{15}{7}\), or we could work it out as follows to see which expression corresponds to our working:
Starting from \(0 = 7m - 15\), if we were to rearrange it as \(7m = 15\), we could divide:
To find an equivalent form for \(m = \frac{15}{7}\), we could also illustrate the steps from rearranging:
The equation could also be illustrated as:
- Adding \(15\) to both sides, we have:
\[
4m + 15 = 3m
\]
However, looking at the expressions provided:
1. **\(4 + 15 - 3\)**: This does not match our rearrangement process.
2. **\(4 - 15 + 3\)**: This does not match our rearrangement process.
3. **\(\frac{-15}{-3 - 4}\)**: This could be viewed as \(-15\) divided by \(-7\), and simplifies to \(m = \frac{15}{7}\) which holds. **(Equivalent)**
4. **\(-15 - (-15) - \frac{-3}{4}\)**: This does not match our rearrangement process.
Thus, the expression that Nathaniel will actually evaluate, based on the transformation to solve \(m\), is:
**\(\frac{-15}{-3 - 4}\)**.
1. Start with the original equation:
\[
-3m = 4m - 15
\]
2. Add \(3m\) to both sides to move all terms involving \(m\) to one side:
\[
0 = 4m + 3m - 15 \quad \Rightarrow \quad 0 = 7m - 15
\]
3. Next, add \(15\) to both sides to isolate the \(m\) term:
\[
15 = 7m
\]
4. Finally, divide both sides by \(7\) to solve for \(m\):
\[
m = \frac{15}{7}
\]
Now, let's express this process using the expressions provided to evaluate.
From our rearrangement, we reached \(m = \frac{15}{7}\), or we could work it out as follows to see which expression corresponds to our working:
Starting from \(0 = 7m - 15\), if we were to rearrange it as \(7m = 15\), we could divide:
To find an equivalent form for \(m = \frac{15}{7}\), we could also illustrate the steps from rearranging:
The equation could also be illustrated as:
- Adding \(15\) to both sides, we have:
\[
4m + 15 = 3m
\]
However, looking at the expressions provided:
1. **\(4 + 15 - 3\)**: This does not match our rearrangement process.
2. **\(4 - 15 + 3\)**: This does not match our rearrangement process.
3. **\(\frac{-15}{-3 - 4}\)**: This could be viewed as \(-15\) divided by \(-7\), and simplifies to \(m = \frac{15}{7}\) which holds. **(Equivalent)**
4. **\(-15 - (-15) - \frac{-3}{4}\)**: This does not match our rearrangement process.
Thus, the expression that Nathaniel will actually evaluate, based on the transformation to solve \(m\), is:
**\(\frac{-15}{-3 - 4}\)**.
-15/-3-4
GPT-4o mini
Let's analyze the expression \(-15 / -3 - 4\) step by step.
1. **Evaluate \(-15 / -3\)**:
\[
-15 / -3 = 5
\]
2. **Subtract \(4\) from that result**:
\[
5 - 4 = 1
\]
So, the final result is:
\[
-15 / -3 - 4 = 1
\]
If we were to relate this back to solving the equation \(-3m = 4m - 15\):
From our earlier solution, we found that:
- The solution for \(m\) is derived from setting up the equation to isolate \(m\) which led to:
\[
7m = 15 \implies m = \frac{15}{7}
\]
However, the evaluation of the expression \(-15 / -3 - 4\) yields a final result of **1**. If you are looking to relate this back to equation manipulation regarding \(m\) as a comparison to find \(m\), remember from the original rearrangement \(7m = 15\) simplifying to \(m = \frac{15}{7}\), reflects another direction entirely.
Let me know if you need any further insights or clarifications!
1. **Evaluate \(-15 / -3\)**:
\[
-15 / -3 = 5
\]
2. **Subtract \(4\) from that result**:
\[
5 - 4 = 1
\]
So, the final result is:
\[
-15 / -3 - 4 = 1
\]
If we were to relate this back to solving the equation \(-3m = 4m - 15\):
From our earlier solution, we found that:
- The solution for \(m\) is derived from setting up the equation to isolate \(m\) which led to:
\[
7m = 15 \implies m = \frac{15}{7}
\]
However, the evaluation of the expression \(-15 / -3 - 4\) yields a final result of **1**. If you are looking to relate this back to equation manipulation regarding \(m\) as a comparison to find \(m\), remember from the original rearrangement \(7m = 15\) simplifying to \(m = \frac{15}{7}\), reflects another direction entirely.
Let me know if you need any further insights or clarifications!