Translating Words to Linear Equations - PSAT Math
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We have three dogs: Joule, Newton, and Toby. Joule is three years older than twice Newton's age. Newton is Toby's age younger than eleven years. Toby is one year younger than Joules age. Find the age of each dog.
We have three dogs: Joule, Newton, and Toby. Joule is three years older than twice Newton's age. Newton is Toby's age younger than eleven years. Toby is one year younger than Joules age. Find the age of each dog.
First, translate the problem into three equations. The statement, "Joule is three years older than twice Newton's age" is mathematically translated as
where 
represents Joule's age and 
is Newton's age.
The statement, "Newton is Toby's age younger than eleven years" is translated as
where 
is Toby's age.
The third statement, "Toby is one year younger than Joule" is
.
So these are our three equations. To figure out the age of these dogs, first I will plug the third equation into the second equation. We get
Plug this equation into the first equation to get
Solve for 
. Add 
to both sides
Divide both sides by 3
So Joules is 9 years old. Plug this value into the third equation to find Toby's age
Toby is 8 years old. Use this value to find Newton's age using the second equation
Now, we have the age of the following dogs:
Joule: 9 years
Newton: 3 years
Toby: 8 years
First, translate the problem into three equations. The statement, "Joule is three years older than twice Newton's age" is mathematically translated as
where
The statement, "Newton is Toby's age younger than eleven years" is translated as
where
The third statement, "Toby is one year younger than Joule" is
So these are our three equations. To figure out the age of these dogs, first I will plug the third equation into the second equation. We get
Plug this equation into the first equation to get
Solve for
Divide both sides by 3
So Joules is 9 years old. Plug this value into the third equation to find Toby's age
Toby is 8 years old. Use this value to find Newton's age using the second equation
Now, we have the age of the following dogs:
Joule: 9 years
Newton: 3 years
Toby: 8 years
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Teachers at an elementary school have devised a system where a student's good behavior earns him or her tokens. Examples of such behavior include sitting quietly in a seat and completing an assignment on time. Jim sits quietly in his seat 2 times and completes assignments 3 times, earning himself 27 tokens. Jessica sits quietly in her seat 9 times and completes 6 assignments, earning herself 69 tokens. How many tokens is each of these two behaviors worth?
Teachers at an elementary school have devised a system where a student's good behavior earns him or her tokens. Examples of such behavior include sitting quietly in a seat and completing an assignment on time. Jim sits quietly in his seat 2 times and completes assignments 3 times, earning himself 27 tokens. Jessica sits quietly in her seat 9 times and completes 6 assignments, earning herself 69 tokens. How many tokens is each of these two behaviors worth?
Since this is a long word problem, it might be easy to confuse the two behaviors and come up with the wrong answer. Let's avoid this problem by turning each behavior into a variable. If we call "sitting quietly" 
and "completing assignments" 
, then we can easily construct a simple system of equations,
and
.
We can multiply the first equation by 
to yield 
.
This allows us to cancel the 
terms when we add the two equations together. We get 
, or 
.
A quick substitution tells us that 
. So, sitting quietly is worth 3 tokens and completing an assignment on time is worth 7.
Since this is a long word problem, it might be easy to confuse the two behaviors and come up with the wrong answer. Let's avoid this problem by turning each behavior into a variable. If we call "sitting quietly"
and
We can multiply the first equation by
This allows us to cancel the
A quick substitution tells us that
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Read, but do not solve, the following problem:
Adult tickets to the zoo sell for $11; child tickets sell for $7. One day, 6,035 tickets were sold, resulting in $50,713 being raised. How many adult and child tickets were sold?
If 
and 
stand for the number of adult and child tickets, respectively, which of the following systems of equations can be used to answer this question?
Read, but do not solve, the following problem:
Adult tickets to the zoo sell for $11; child tickets sell for $7. One day, 6,035 tickets were sold, resulting in $50,713 being raised. How many adult and child tickets were sold?
If
6,035 total tickets were sold, and the total number of tickets is the sum of the adult and child tickets, 
.
Therefore, we can say 
.
The amount of money raised from adult tickets is $11 per ticket mutiplied by 
tickets, or 
dollars; similarly, 
dollars are raised from child tickets. Add these together to get the total amount of money raised:
These two equations form our system of equations.
6,035 total tickets were sold, and the total number of tickets is the sum of the adult and child tickets,
Therefore, we can say
The amount of money raised from adult tickets is $11 per ticket mutiplied by
These two equations form our system of equations.
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Solve the following story problem:
Jack and Aaron go to the sporting goods store. Jack buys a glove for 
and 
wiffle bats for 
each. Jack has 
left over. Aaron spends all his money on 
hats for 
each and 
jerseys. Aaron started with 
more than Jack. How much does one jersey cost?
Solve the following story problem:
Jack and Aaron go to the sporting goods store. Jack buys a glove for
Let's call "
" the cost of one jersey (this is the value we want to find)
Let's call the amount of money Jack starts with "
"
Let's call the amount of money Aaron starts with "
"
We know Jack buys a glove for 
and 
bats for 
each, and then has 
left over after. Thus:
simplifying, 
so Jack started with 
We know Aaron buys 
hats for 
each and 
jerseys (unknown cost "
") and spends all his money.
The last important piece of information from the problem is Aaron starts with 
dollars more than Jack. So:
From before we know:
Plugging in:
so Aaron started with 
Finally we plug 
into our original equation for A and solve for x:
Thus one jersey costs 
Let's call "
Let's call the amount of money Jack starts with "
Let's call the amount of money Aaron starts with "
We know Jack buys a glove for
simplifying,
We know Aaron buys
The last important piece of information from the problem is Aaron starts with
From before we know:
Plugging in:
so Aaron started with
Finally we plug
Thus one jersey costs
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