How many cans does each small box hold?
Understand the Problem
The question is asking how many cans each small box holds, given the total number of cans filled in both small and large boxes. We need to set up a system of equations to solve for the unknowns.
Answer
Each small box holds \( 16 \) cans.
Answer for screen readers
Each small box holds ( 16 ) cans.
Steps to Solve
- Define Variables
Let ( x ) be the number of cans in each large box, and ( y ) be the number of cans in each small box.
- Set Up the First Equation
From the information given, we know that Estelle fills 3 large boxes and 5 small boxes with a total of 170 cans. This gives us the equation: $$ 3x + 5y = 170 $$
- Set Up the Second Equation
The other boy fills 4 large boxes and 4 small boxes with a total of 184 cans. This leads to the equation: $$ 4x + 4y = 184 $$
- Simplify the Second Equation
We can simplify the second equation by dividing every term by 4: $$ x + y = 46 $$
- Substitute into First Equation
Now we substitute ( y ) from the simplified equation ( y = 46 - x ) into the first equation: $$ 3x + 5(46 - x) = 170 $$
- Solve for ( x )
Distribute and combine like terms: $$ 3x + 230 - 5x = 170 $$ $$ -2x + 230 = 170 $$ Subtract 230 from both sides: $$ -2x = -60 $$ Now divide by -2: $$ x = 30 $$
- Find ( y )
Now we can find ( y ) using the equation ( y = 46 - x ): $$ y = 46 - 30 $$ $$ y = 16 $$
- Conclusion
Each small box holds 16 cans.
Each small box holds ( 16 ) cans.
More Information
Estelle's situation gives us a practical example of using systems of equations to solve real-world problems. By breaking down the total numbers into manageable equations, we can easily find the unknown variables.
Tips
- Mixing up the equations when setting them up.
- Forgetting to simplify equations properly.
- Miscalculating values during substitution and combining like terms.
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