A nut store normally sells cashews for 9.00 per pound and...

A nut store normally sells cashews for $\$ 9.00$ per pound and almonds for $\$ 3.50$ per pound. But at the end of the month the almonds had not sold well, so, in order to sell 60 pounds of almonds, the manager decided to $\operatorname{mix}$ the 60 pounds of almonds with some cashews and sell the mixture for $\$ 7.50$ per pound. How many pounds of cashews should be mixed with the almonds to ensure no change in the revenue?

Key Concepts

Mixture Problem

A mixture problem involves combining two or more components with different characteristics, such as cost or concentration, to obtain a mixture that meets a specific criterion, like a desired average property or total revenue. It typically requires determining the correct amount of each component to reach the intended goal.

Weighted Average

The weighted average is a method for finding the average value of a mixture when different parts contribute unequally. In mixture problems, each component's contribution is proportional to its quantity, and the overall average is calculated by taking into account both the individual values and their corresponding weights.

Linear Equations

Linear equations are used to represent the relationships between the quantities and values involved in a mixture. By setting up an equation that equates the total revenue or average value of the mixture to the combined contributions of its components, one can solve for the unknown quantities, ensuring that the mixture meets the desired condition.

Transcript

00:01 As we begin this problem, just remember, then order to find the total amount of money of something that's being sold, we do the rate, so the cost per pound, times the amount, so in this case the number of pounds.

00:13 This will help us to solve this problem.

00:15 So we have cash use, and they're $9 a pound, and we have almonds and they're $350 a pound.

00:22 We want to mix them together so that we get $7 .50 per pound.

00:26 That means what we need to change right here is the amount.

00:29 What we're looking for is the amount of cash use.

00:32 So we've been told that we're going to be mixing in 60 pounds of almonds.

00:38 Let's come up with a variable for cash use, because this is what we're aiming for.

00:41 Let's call it x.

00:43 So that means the total amount of money that we have in cashews is 9 times x.

00:50 And we're adding that to the total amount of money in almonds, which is going to be 350 times 60.

00:58 Because again, for that, we know the amount.

01:03 Now we want this to be equal to what's going on on the other side.

01:07 At the end of this, we know we want it to cost $7 .50 as a rate, and that means we just need to figure out the amount.

01:15 Well, we know that we're going to have x pounds of cashews plus 60 pounds of almonds.

01:24 Now, we want to distribute here, because we're going to be getting $350 ,000 times x plus 60.

01:34 And i see parentheses.

01:35 So let's distribute that 750 to the x and to the 60.

01:41 That means on this other side of the equation, we have 450 because 7 .5 times 60 is 450 plus 7 .5x...

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