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Garden A produces $5.8$ times as many vegetables as garden B. Together the gardens produce 102 pounds of vegetables. How many pounds of vegetables does garden A produce?
Garden A produces 87 pounds of vegetables.
Two consecutive even numbers sum to 432. What are the two numbers?
$\begin{array}{ll}\text{Step}\text{\hspace{0.17em}}\text{1:}\hfill & \text{Let}\text{\hspace{0.17em}}x=\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{smaller}\text{\hspace{0.17em}}\text{even}\text{\hspace{0.17em}}\text{number}\text{.}\text{\hspace{0.17em}}\text{Then}\text{\hspace{0.17em}}x+2=\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{next}\text{\hspace{0.17em}}\text{(consecutive)}\text{\hspace{0.17em}}\text{even}\text{\hspace{0.17em}}\text{number}\hfill \\ \hfill & \text{since}\text{\hspace{0.17em}}\text{consecutive}\text{\hspace{0.17em}}\text{even}\text{\hspace{0.17em}}\text{numbers}\text{\hspace{0.17em}}\text{differ}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{2}\text{\hspace{0.17em}}\text{(as}\text{\hspace{0.17em}}\text{do}\text{\hspace{0.17em}}\text{consecutive}\text{\hspace{0.17em}}\text{odd}\text{\hspace{0.17em}}\text{numbers)}\text{.}\hfill \end{array}$
$\begin{array}{lllll}\text{Step}\text{\hspace{0.17em}}\text{2:}\hfill & \hfill x+x+2& =\hfill & 432.\hfill & \hfill \\ \text{Step}\text{\hspace{0.17em}}\text{3:}\hfill & \hfill x+x+2& =\hfill & 432\hfill & \hfill \\ \hfill & \hfill 2x+2& =\hfill & 432\hfill & \hfill \\ \hfill & \hfill 2x& =\hfill & 430\hfill & \hfill \\ \hfill & \hfill x& =\hfill & 215.\hfill & \text{Also,}\text{\hspace{0.17em}}\text{since}\text{\hspace{0.17em}}x=215,\text{\hspace{0.17em}}x+2=217.\hfill \end{array}$
$\begin{array}{ll}\text{Step}\text{\hspace{0.17em}}4:\hfill & 215+217=432,\text{\hspace{0.17em}}\text{but}\text{\hspace{0.17em}}215\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}217\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{odd}\text{\hspace{0.17em}}\text{numbers}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{we}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{looking}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{even}\text{\hspace{0.17em}}\text{numbers}\text{.}\hfill \\ \hfill & \text{Upon}\text{\hspace{0.17em}}\text{checking}\text{\hspace{0.17em}}\text{our}\text{\hspace{0.17em}}\text{method}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{solution}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{reexamining}\text{\hspace{0.17em}}\text{our}\text{\hspace{0.17em}}\text{equation},\text{\hspace{0.17em}}\text{we}\text{\hspace{0.17em}}\text{find}\text{\hspace{0.17em}}\text{no}\hfill \\ \hfill & \text{mistakes}\text{.}\hfill \\ \text{Step}\text{\hspace{0.17em}}5:\hfill & \text{We}\text{\hspace{0.17em}}\text{must}\text{\hspace{0.17em}}\text{conclude}\text{\hspace{0.17em}}\text{that}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{problem}\text{\hspace{0.17em}}\text{has}\text{\hspace{0.17em}}\text{no}\text{\hspace{0.17em}}\text{solution}\text{.}\text{\hspace{0.17em}}\text{There}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{no}\text{\hspace{0.17em}}\text{two}\text{\hspace{0.17em}}\text{consecutive}\text{\hspace{0.17em}}even\hfill \\ \hfill & \text{numbers}\text{\hspace{0.17em}}\text{that}\text{\hspace{0.17em}}\text{sum}\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}\text{432}\text{.}\hfill \end{array}$
The sum of two consecutive even numbers is 498. What are the two numbers?
The two numbers are 248 and 250.
Solve the following problems. Note that some of the problems may seem to have no practical applications and may not seem very interesting. They, along with the other problems, will, however, help to develop your logic and problem-solving ability.
If eighteen is subtracted from some number the result is fifty-two. What is the number?
Step 1: Let
$x$ = the unknown quantity.
Step 2: The equation is
$x-18=52.$
Step 3: (Solve the equation.) Add 18 to each side.
$x-18+18=52+18$
$x=70$
Step 4: (Check)
$70-18=52;$ True.
Step 5: The number is 70.
If nine more than twice a number is forty-six, what is the number?
If nine less than three eighths of a number is two and one fourth, what is the number?
Step 5: The number is 30.
Twenty percent of a number is 68. What is the number?
Eight more than a quantity is 37. What is the original quantity?
Step 5: The original quantity is 29.
If a quantity plus $85\%$ more of the quantity is $62.9$ , what is the original quantity?
A company must increase production by $12\%$ over last year’s production. The new output will be 56 items. What was last year’s output?
Step 5: Last year's output was 50 items.
A company has determined that it must increase production of a certain line of goods by $1\frac{1}{2}$ times last year’s production. The new output will be 2885 items. What was last year’s output?
A proton is about 1837 times as heavy as an electron. If an electron weighs $2.68$ units, how many units does a proton weigh?
Step 5: A proton weighs $\text{4923}\text{.16}$ units.
Neptune is about 30 times as far from the sun as is the Earth. If it takes light 8 minutes to travel from the sun to the Earth, how many minutes does it take to travel to Neptune?
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