Raspberry and Grape Mixture Problem: Two-Variable System

What This Problem Teaches

• Setting up and solving systems of two linear equations with real-world constraints
• Translating word problems into mathematical relationships using quantity and value equations
• Working strategically with decimal coefficients by clearing them early in the solution
• Recognizing mixture problems as a fundamental application of linear systems
• Checking solutions by verifying they satisfy both physical and financial constraints

Solution: The Elimination Method

Step 1 — Define the variables

Let r = pounds of raspberries sold and g = pounds of grapes sold.

Step 2 — Set up the system of equations

We need two equations based on the given information:

Step 3 — Eliminate decimals

To avoid calculation errors, multiply the second equation by 100:

Step 4 — Use elimination to solve

Multiply the first equation by 250 to match the coefficient of r:

Subtract the first equation from the second:

Step 5 — Find the other variable

Substitute g = 52.2 into the first equation:

Solution: Method 2 — Substitution Approach

Step 1 — Express one variable in terms of the other

From the weight equation r + g = 66, we get:

Step 2 — Substitute into the revenue equation

Replace r in the revenue equation:

Step 3 — Solve for g

Distribute and combine like terms:

Step 4 — Find r using substitution

Verification

Let's check our solution by substituting back into both original equations:

Weight check:13.8 + 52.2 = 66 ✓

Revenue check:

Both conditions are satisfied, confirming our solution is correct.

Watch Out For These

The Underlying Pattern

Every two-item mixture problem follows the same structure. If you have quantities x and y with unit values v₁ and v₂, total quantity Q, and total value V, then:

The key insight is that you always need exactly two pieces of information to find two unknowns. Mixture problems provide total quantity and total value, which gives you the perfect setup for a linear system.

How to Spot This Problem Type

Look for these tell-tale phrases that signal a mixture problem:

• "Each [item A] costs $X and each [item B] costs $Y"
• "Total of [number] pounds/gallons/units"
• "Made $[amount] from selling" or "total revenue/value was"
• "Mixture," "combination," or "blend" of two different items with different prices
• Problems involving coffee blends, fruit sales, ticket sales with different prices, or investments with different rates

The pattern is always: two different items, different unit prices, known total quantity, and known total value. Once you recognize this structure, you know you're setting up the same type of system every time.

Does This Seem Reasonable?

Let's do a quick sanity check on our answer:

Arun sold 13.8 pounds of raspberries and 52.2 pounds of grapes. Notice that he sold almost four times as many grapes as raspberries. Does this make sense?

Since grapes cost $3.75/lb and raspberries cost $2.50/lb, grapes are 50% more expensive. If Arun had sold equal amounts of each (33 lbs each), the revenue would have been:

But he actually made $230.25, which is $24 more than this "equal split" scenario. To make that extra money, he needed to sell more of the expensive grapes and fewer of the cheap raspberries. So yes, the lopsided split toward grapes makes perfect economic sense.

Where This Shows Up in Real Life

• Coffee roasting: Blending different coffee beans with different costs per pound to achieve a target blend price and total quantity
• Investment portfolios: Allocating money between investments with different returns when you know the total invested and total return
• Manufacturing: Combining raw materials with different costs to produce a fixed quantity at a target total material cost

What If?