The mass of a jar is 750 g when it is 1/2 filled with orange juice. The sand jar has a mass of 625 g when it is 1/4 filled with oranges.
(a) What is the mass of the jar when it is 3/4 filled with orange juice?
(b) What is the mass of the jar when it is empty?
To solve this problem, we can set up a system of equations. Let's define some variables:
Let's assume the mass of the empty jar is J grams.
Let X be the mass of orange juice in grams that fills 1/4 of the jar.
Let Y be the mass of oranges in grams that fills 1/4 of the jar.
From the given information, we can set up the following equations:
Equation 1: J + X = 750 (1) (J = mass of jar when it is 1/2 filled with orange juice)
Equation 2: J + Y = 625 (2) (J = mass of jar when it is 1/4 filled with oranges)
Now, let's solve these equations step-by-step to find the answers.
(a) What is the mass of the jar when it is 3/4 filled with orange juice?
To find the mass of the jar when it is 3/4 filled with orange juice, we need to find the mass of orange juice that fills 3/4 of the jar and add it to the mass of the empty jar.
Let's assume the mass of the orange juice that fills 3/4 of the jar is Z grams.
Equation 3: J + 3Z = ? (3) (J = mass of jar when it is 3/4 filled with orange juice)
To find Z, we can use the fact that the mass of orange juice doubles when the jar is filled halfway:
J + X = 750
Since X is the mass of orange juice that fills 1/4 of the jar, it follows that 2X is the mass of orange juice that fills 1/2 of the jar.
2X = J
Simplifying this equation, we have:
X = J/2
Substituting this value of X in Equation 1, we have:
J + J/2 = 750
Combining like terms, we get:
3J/2 = 750
To solve for J, multiply both sides of the equation by 2/3:
J = (750 * 2)/3
J = 500
So, the mass of the empty jar is 500 grams.
Now, we can substitute the value of J in Equation 3:
J + 3Z = 500 + 3Z = ?
Solving this equation step-by-step will give us the answer:
500 + 3Z = ?
First, we need to find the value of Z. To do that, we need another equation.
Using the fact that the mass of the jar when it is 1/4 filled with oranges is 625 grams, we can set up the following equation:
J + Y = 625
Substituting the value of J (500) and rearranging the equation, we have:
500 + Y = 625
Subtracting 500 from both sides of the equation, we get:
Y = 625 - 500
Y = 125
Now, we have Y = 125, X = J/2 = 500/2 = 250, and J = 500.
Substituting these values in Equation 2, we have:
J + Y = 625
500 + 125 = 625
The values of X, Y, and J satisfy this equation, which indicates that our assumptions are correct.
Now, let's go back to Equation 3, where we substitute J = 500:
500 + 3Z = ?
Subtracting 500 from both sides of the equation gives us:
3Z = ?
Now, by dividing both sides of the equation by 3, we solve for Z:
Z = ?/3
To find the mass of the jar when it is 3/4 filled with orange juice, we need to find the value of Z. However, at this point, we don't have enough information to determine Z. The problem statement does not provide the necessary information to calculate the mass of the jar when it is 3/4 filled with orange juice.
(b) What is the mass of the jar when it is empty?
From our previous calculations, we found that the mass of the empty jar is 500 grams. Therefore, the mass of the jar when it is empty is 500 grams.
To solve these problems, we need to determine the mass of the jar and the mass of the liquid inside it. Then we can use that information to find the mass when it is partially filled.
Let's break down the given information:
The mass of the jar when it is 1/2 filled with orange juice is 750 g.
The mass of the jar when it is 1/4 filled with oranges is 625 g.
(a) What is the mass of the jar when it is 3/4 filled with orange juice?
To find the mass of the jar when it is 3/4 filled with orange juice, we first need to determine the mass of the liquid itself. From the given information, we know that when the jar is 1/2 filled with orange juice, its mass is 750 g.
Since the liquid (orange juice) is the only thing that changes when the jar is filled, we can calculate the mass of the liquid when it is 3/4 filled using proportions.
Let's set up a proportion:
(750 g)/(1/2) = (x g)/(3/4)
To solve for x (the mass of the liquid when the jar is 3/4 filled), we cross-multiply and solve for x:
750 g * (3/4) = x g
x = 562.5 g
Now, to find the total mass of the jar when it is 3/4 filled, we need to add the mass of the jar itself to the mass of the liquid:
Total mass = mass of jar + mass of liquid
Total mass = 750 g + 562.5 g
Total mass = 1312.5 g
Therefore, the mass of the jar when it is 3/4 filled with orange juice is 1312.5 g.
(b) What is the mass of the jar when it is empty?
To find the mass of the jar when it is empty, we can set up a similar proportion:
(625 g)/(1/4) = (x g)/(0)
Since the jar is empty, the mass of the liquid is zero. We can simplify the proportion to:
625 g/(1/4) = x g/0
Since we cannot divide by zero, this tells us that the mass of the jar when it is empty is undefined or not meaningful.
I assume you mean orange juice in both cases.
Then if the empty jar's mass is m, and the juice's mass when its full is j, then we have
m + 1/2 j = 750
m + 1/4 j = 625
m = 500 g
m + 3/4 j = 750+625-500 = 825 g