Priya is buying raisins and almonds to make trail mix. Almonds cost $5.20 per pound and raisins cost $2.75 per pound. Priya spent $11.70 buying almonds and raisins. How many pounds... Priya is buying raisins and almonds to make trail mix. Almonds cost $5.20 per pound and raisins cost $2.75 per pound. Priya spent $11.70 buying almonds and raisins. How many pounds of raisins did Priya buy if she bought the following amounts of almonds? 2 pounds, 1.06 pounds, 0.64 pounds, and a pounds of almonds? Also, solve the linear equation 2x + 4y - 31 = 123 for x and y.
Understand the Problem
The question contains mathematical problems related to buying almonds and raisins, including how many pounds of raisins Priya could buy depending on the pounds of almonds, and solving a linear equation with two variables. The first part involves applying a cost equation, and the second part is about solving for variables x and y in the given equation.
Answer
(a) $1.30$; (b) $4.25$; (c) $6.04$; (d) $r = \frac{11.70 - 5.20a}{2.75}$; Linear equations: $x = 77 - 2y$, $y = \frac{154 - 2x}{4}$.
Answer for screen readers
(a) Priya can buy $1.30$ pounds of raisins with 2 pounds of almonds.
(b) Priya can buy $4.25$ pounds of raisins with 1.06 pounds of almonds.
(c) Priya can buy $6.04$ pounds of raisins with 0.64 pounds of almonds.
(d) The general formula for $r$ is $$ r = \frac{11.70 - 5.20a}{2.75} $$.
For the linear equations, $$ x = 77 - 2y $$ and $$ y = \frac{154 - 2x}{4} $$.
Steps to Solve
-
Understanding the Cost Equation Priya’s total cost for almonds and raisins is given by the equation $$ 5.20a + 2.75r = 11.70 $$ where $a$ is the pounds of almonds and $r$ is the pounds of raisins. We will solve for the amount of raisins $r$ for various values of $a$.
-
Substituting Values for Almonds For each given amount of almonds ($a$), substitute it into the cost equation to find $r$. The equation rearranged to find $r$ is: $$ r = \frac{11.70 - 5.20a}{2.75} $$
-
Calculating Raisins for Each Almond Amount For each option:
-
(a) $2$ pounds of almonds: $$ r = \frac{11.70 - 5.20 \cdot 2}{2.75} $$ Calculate this.
-
(b) $1.06$ pounds of almonds: $$ r = \frac{11.70 - 5.20 \cdot 1.06}{2.75} $$ Calculate this.
-
(c) $0.64$ pounds of almonds: $$ r = \frac{11.70 - 5.20 \cdot 0.64}{2.75} $$ Calculate this.
-
(d) $a$ pounds of almonds: $$ r = \frac{11.70 - 5.20a}{2.75} $$ This is the general form to find $r$ based on any number of pounds of almonds.
-
Solving the Linear Equation For the second problem involving the linear equation: $$ 2x + 4y - 31 = 123 $$ Rearranging gives: $$ 2x + 4y = 154 $$
-
Expressing x in terms of y Divide everything by 2: $$ x + 2y = 77 $$ Now: $$ x = 77 - 2y $$
-
Expressing y in terms of x From $$ 2x + 4y = 154 $$ We can isolate $y$: $$ 4y = 154 - 2x $$ $$ y = \frac{154 - 2x}{4} $$
(a) Priya can buy $1.30$ pounds of raisins with 2 pounds of almonds.
(b) Priya can buy $4.25$ pounds of raisins with 1.06 pounds of almonds.
(c) Priya can buy $6.04$ pounds of raisins with 0.64 pounds of almonds.
(d) The general formula for $r$ is $$ r = \frac{11.70 - 5.20a}{2.75} $$.
For the linear equations, $$ x = 77 - 2y $$ and $$ y = \frac{154 - 2x}{4} $$.
More Information
The calculations show how the amounts of raisins depend on the specific amounts of almonds Priya buys. This also illustrates how to rearrange and solve linear equations with two variables using simple algebra.
Tips
- Confusing the costs per pound with the total cost. Carefully pay attention to units and ensure you’re using the right equation.
- Not rearranging equations properly. Double-check each step when isolating variables.
AI-generated content may contain errors. Please verify critical information
