At an animal shelter, the number of dog adoptions one weekend was 10 less than 3 times the number of cat adoptions. The number of cat adoptions plus twice the number of dog adoptio... At an animal shelter, the number of dog adoptions one weekend was 10 less than 3 times the number of cat adoptions. The number of cat adoptions plus twice the number of dog adoptions was 8. How many cats and how many dogs were adopted that weekend?
Understand the Problem
The question is a word problem involving two variables: the number of cat adoptions and the number of dog adoptions. We have two equations based on the information provided: (1) the number of dog adoptions is 10 less than 3 times the number of cat adoptions, and (2) the number of cat adoptions plus twice the number of dog adoptions is 8. We need to solve this system of equations to find the values for both variables.
Answer
The number of cat adoptions is 4 and the number of dog adoptions is 2.
Answer for screen readers
There were 4 cat adoptions and 2 dog adoptions.
Steps to Solve
- Define variables
Let $c$ be the number of cat adoptions and $d$ be the number of dog adoptions.
- Write the first equation
"The number of dog adoptions one weekend was 10 less than 3 times the number of cat adoptions." can be written as:
$d = 3c - 10$
- Write the second equation
"The number of cat adoptions plus twice the number of dog adoptions was 8" can be written as:
$c + 2d = 8$
- Solve the system of equations
Substitute the first equation into the second equation:
$c + 2(3c - 10) = 8$
- Simplify and solve for $c$
Expand and simplify the equation:
$c + 6c - 20 = 8$
$7c - 20 = 8$
$7c = 28$
$c = 4$
- Solve for $d$
Substitute the value of $c$ back into the first equation:
$d = 3(4) - 10$
$d = 12 - 10$
$d = 2$
There were 4 cat adoptions and 2 dog adoptions.
More Information
This problem demonstrates how a system of equations can be used to solve real-world problems. By defining the variables and setting up the equations correctly, we can find the values of the unknowns.
Tips
A common mistake is to misinterpret the wording of the word problem and set up the equations incorrectly. For example, "10 less than 3 times the number of cat adoptions" may be misinterpreted as $3(c-10)$ instead of $3c - 10$. Careful reading and translating the words into mathematical expressions can help avoid this mistake. Another common mistake is in the algebraic manipulation.
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