Build and use a formula in two unknowns

Juice box

\(\displaystyle 90\text{ cents}\)

Roll

\(\displaystyle \$3.50\)

Juice boxes bought

\(\displaystyle 25\)

Rolls bought

\(\displaystyle 12\)

A canteen sells juice boxes at 90 cents each and rolls at $3.50 each. A class buys 25 juice boxes and 12 rolls. What is the total cost?

Rows n and p mean notebooks and pens. Change the notebook and pen counts. Test zero of both, equal nonzero counts, and unequal counts. Compare the true total with the product of the two sums

When did the total cost equal the sum of the rates times the sum of the counts?

A formula with two unknowns

A total made from two item types has one rate times one count for each type. Keep both rates in the same money unit, then add the two costs

1. 1Define the variables Use one variable for the number of each item \(\displaystyle \begin{gathered}j=\text{juice-box count}\\r=\text{roll count}\end{gathered}\)
2. 2Match each rate Multiply each unit rate by the count of its own item \(\displaystyle C=0.90j+3.50r\)
3. 3Use one money unit Convert a cents rate to dollars before combining it with a dollar rate \(\displaystyle 90\ \text{cents}=\$0.90\)
4. 4Add the two costs For 25 juice boxes and 12 rolls, add the two item costs to find the total value \(\displaystyle \begin{aligned}C&=0.90(25)+3.50(12)\\&=22.50+42\\&=64.50\end{aligned}\)

Raffle ticket

\(\displaystyle 50\text{ cents}\)

Hot dog

\(\displaystyle \$4.20\)

Tickets sold

\(\displaystyle 140\)

Hot dogs sold

\(\displaystyle 65\)

A fundraising stall sells raffle tickets at 50 cents each and hot dogs at $4.20 each. The stall sold 140 tickets and 65 hot dogs. What were the stall's takings?

Convert 50 cents to dollars first, then match each rate to its own count before adding

Raffle ticket

\(\displaystyle 50\text{ cents}\)

Hot dog

\(\displaystyle \$4.20\)

Second stall tickets sold

\(\displaystyle 90\)

Second stall takings

\(\displaystyle \$246.60\)

A second stall sold 90 raffle tickets and some hot dogs, with total takings of $246.60. How many hot dogs did the second stall sell?

Take the ticket money off the takings first, then divide what is left by the hot dog rate

Juice box

\(\displaystyle 90\text{ cents}\)

Roll

\(\displaystyle \$3.50\)

Swapping the unit rates. Let j be the number of juice boxes and r the number of rolls. One of these lines is right. Which one?

Whole non-negative counts n and p satisfy 2n+p=10. Which statement is true?

Equations

\(\displaystyle \begin{gathered}2n+p=10\\n+p=7\end{gathered}\)

Now the same two counts also total 7. Find n.