Two-Step Equations Word Problems Practice

1. Cookies are on sale! Today each cookie costs ‍$0.75 less than the normal price. Right now if you buy ‍7 of them it will only cost you $2.80!

   Write an equation to determine the normal price of each cookie $$(c)$$.

   Find the normal price of each cookie.

2. John read the first ‍114 pages of a novel, which was ‍3 pages less than ⅓ of the novel.

   Write an equation to determine the total number of pages $$(p)$$ in the novel.

   Find the total number of pages in the novel.

3. In winter, the price of apples suddenly went up by ‍$0.75 per pound. Sam bought ‍3 pounds of apples at the new price for a total of ‍$5.88.

   Write an equation to determine the original price per pound $$(p)$$.

   Find the original price per pound.

4. The perimeter of a rectangle is ‍34 units. Its width is ‍6.5 units.

   Write an equation to determine the length ‍$$(l)$$ of the rectangle.

   Find the length of the rectangle.

Source: Khan Academy, https://www.khanacademy.org/math/algebra-home/alg-basic-eq-ineq/alg-linear-eq-word-probs/e/linear-equation-world-problems-2
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

1. 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 ‍$$\$1.15$$.

2. 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.

3. 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.
   

4. 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.