Johanna bought $12$ items at the college bookstore. The items cost a total of $\$ 29.00$ . The pens cost $\$ 0.50$ each, the notebooks were $\$ 4.00$ each, and the highlighters cost $\$ 1.50$ each. She bought $4$ more notebooks than highlighters. How many of each item did she buy? $\newline$ Use a system of three linear equations to solve the problem.

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Grade 6

Divide fractions and mixed numbers: word problems

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Q. Johanna bought

12

items at the college bookstore. The items cost a total of

\$ 29.00

. The pens cost

\$ 0.50

each, the notebooks were

\$ 4.00

each, and the highlighters cost

\$ 1.50

each. She bought

4

more notebooks than highlighters. How many of each item did she buy?

\newline

Use a system of three linear equations to solve the problem.

Define variables: Let's define the variables: let $p$ be the number of pens, $n$ be the number of notebooks, and $h$ be the number of highlighters Johanna bought.
Write first equation: We know the total number of items bought is $12$ , so we write the first equation: $p + n + h = 12$ .
Write second equation: The total cost of all items is $\$29.00$ . Each pen costs $\$0.50$ , each notebook costs $\$4.00$ , and each highlighter costs $\$1.50$ . We write the second equation based on the total cost: $0.50p + 4n + 1.50h = 29$ .
Write third equation: Johanna bought $4$ more notebooks than highlighters, so we have the third equation: $n = h + 4$ .
Substitute third equation: Substitute the third equation into the first and second equations. Replace $n$ with $h + 4$ in the first equation: $p + (h + 4) + h = 12$ . Simplify to get $p + 2h + 4 = 12$ .
Solve for $p$ : Solve for $p$ : $p = 12 - 2h - 4$ . Simplify further to get $p = 8 - 2h$ .
Substitute $n$ into cost equation: Now substitute $n = h + 4$ into the cost equation: $0.50p + 4(h + 4) + 1.50h = 29$ . This simplifies to $0.50p + 4h + 16 + 1.50h = 29$ .
Substitute $p$ into cost equation: Substitute $p = 8 - 2h$ into the equation: $0.50(8 - 2h) + 4h + 16 + 1.50h = 29$ . Simplify to get $4 - h + 4h + 16 + 1.50h = 29$ .
Combine like terms: Combine like terms: $4 + 5.50h + 16 = 29$ . Simplify to get $5.50h + 20 = 29$ .
Solve for h: Solve for h: $5.50h = 29 - 20$ . Simplify to get $5.50h = 9$ .
Solve for h: Solve for $h$ : $5.50h = 29 - 20$ . Simplify to get $5.50h = 9$ . Divide by $5.50$ to find $h$ : $h = \frac{9}{5.50}$ . This gives $h = 1.64$ , which doesn't make sense because the number of highlighters should be a whole number.