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?

Question

GPT-4o mini · Accepted Answer

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. = .

GPT-4o mini · Answer

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**.

Answer

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 · Answer

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.

Answer

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

Answer

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

GPT-4o mini · Answer

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 $:

Multiply both sides by $ 2 $:

\[ 2a = (b_1 + b_2)h \]
Now, divide both sides by $ h $:

\[ \frac{2a}{h} = b_1 + b_2 \]
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:

\[ 2a = (b_1 + b_2)h \]

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:

$ \frac{2a}{h} - b_2 = b_1 $: This is equivalent. ✔
$ \frac{a}{2h} - b_2 = b_1 $: This is not equivalent (does not match the steps). ✘
$ \frac{2a - b_2}{h} = b_1 $: This is also not equivalent. ✘
$ \frac{2a}{b_1 + b_2} = h $: This is equivalent. ✔
$ \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.

Answer

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.

GPT-4o mini · Answer

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: **$\frac{-15}{-3 - 4}$**.

Answer

-15/-3-4

GPT-4o mini · Answer

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!

Rearranging for \( b_1 \):

Rearranging for \( h \):

Evaluating the Options:

Summary of Equivalent Equations: