Answers

Practice Problems

Let ‍$$c$$ be the normal price of each cookie.

The current price for each cookie is ‍$$c-0.75$$.

The current cost of ‍7 cookies is ‍$$7(c-0.75)$$.

Since the current cost of ‍7 cookies is ‍$2.80, let's set this equal to ‍2.8:

$$7(c-0.75)=2.8$$

Now, let's solve the equation to find the normal price of each cookie (c).

$$\begin{aligned}7(c-0.75)&=2.8\\&\\
\dfrac{7(c-0.75)}{{7}}&=\dfrac{2.8}{{7}}&&\text{divide both sides by ${7}$}\\
\\
c-0.75&=0.4\\
\\
c-{0.75}{+0.75}&=0.4{+0.75}&&{\text{add }} {0.75} \text{ to both sides}\\
\\
c&=1.15\end{aligned}$$

The equation is ‍$$7(c-0.75)=2.8$$.

The normal price of each cookie is ‍$$\$1.15$$.
Let ‍$$p$$ be the total number of pages in the novel.

We can represent $$\dfrac{1}{3}$$ of the novel as $$\dfrac{1}{3}p$$. John has read ‍3 fewer pages than $$\dfrac{1}{3}p$$.

He has read $$\dfrac{1}{3}p-3$$pages.

Since he has read 114 pages, let's set this equal to 114:

Now, let's solve the equation to find the total number of pages $$(p)$$ in the novel.

$$\begin{aligned}
\dfrac{1}3p-3&=114\\
\\
\dfrac{1}3p-3{+3}&=114{+3}&&{\text{add }3} \text{ to each side}\\
\\
\dfrac{1}3p&=117\\
\\
\dfrac{\dfrac{1}3p}{{\dfrac{1}3}}&=\dfrac{117}{{\dfrac{1}3}}&&\text{divide each side by ${\dfrac{1}3}$}\\
\\
p&=351\end{aligned}$$

The equation is $$\dfrac{1}{3}p-3 = 114$$.

The novel has a total of ‍351 pages.
Let $$p$$ be the original price per pound of apples.

The new price is $$p+{0.75}$$ dollars per pound. Sam bought 3 pounds of apples.

Sam's total cost was $$3(p+0.75)$$.

Since his total cost was $$\$5.88$$, let's set this equal to 5.88:

Now, let's solve the equation to find the original price per pound $$(p)$$.

$$\begin{aligned}3(p+0.75)&=5.88\\&\\
\dfrac{3(p+{0.75})}{3}&=\dfrac{5.88}{3}&&\text{divide both sides by $3$}\\
\\
p+{0.75}&=1.96\\
\\
p+{0.75}{-0.75}&=1.96{-0.75}&&{\text{subtract }} {0.75} \text{ from both sides}\\
\\
p&=1.21\end{aligned}$$

The equation is $$3(p+0.75)=5.88$$.

The original price of the apples was $1.21 per pound.
Let $$l$$ be the length of the rectangle.

The perimeter is equal to $$2l+2w$$. Let's substitute in the width of 6.5:

$$\qquad\begin{aligned}&2l+2w\\
=&2l+2(6.5)\\
=&2l+13\end{aligned}$$

The perimeter of the rectangle is $$2l+{13}$$.

Since the perimeter equals 34 units, let's set this equal to 34:

Now, let's solve the equation to find the length of the rectangle $$(l)$$.

$$\begin{aligned}
2l+13&=34\\
\\
2l+13{-13}&=34{-13}&&{\text{subtract }13} \text{ from each side}\\
\\
2l&=21\\
\\
\dfrac{2l}{{2}}&=\dfrac{21}{{2}}&&\text{divide each side by ${2}$}\\
\\
l&=10.5\end{aligned}$$

The equation is $$2l+13=34$$.

The length of the rectangle is 10.5 units.