1Cademy - Solving a Mixture Word Problem: Trail Mix (Raisins and Nuts)

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Solving a Mixture Word Problem: Trail Mix (Raisins and Nuts)

Apply the seven-step problem-solving strategy, the mixture model, and the known-total technique to a mixture word problem involving food products measured by weight.

Problem: Henning is mixing raisins and nuts to make $25$ pounds of trail mix. Raisins cost $4.50 a pound and nuts cost $8 a pound. If Henning wants his cost for the trail mix to be $6.60 a pound, how many pounds of raisins and how many pounds of nuts should he use?

Read the problem and identify the two ingredient types: raisins (worth $4.50 per pound) and nuts (worth $8 per pound). The total mixture is $25$ pounds at a target cost of $6.60 per pound.
Identify what to find: the number of pounds of raisins and the number of pounds of nuts.
Name the unknowns using a single variable. Because the total weight is known, use the known-total technique. Let $x =$ the number of pounds of raisins. Then the number of pounds of nuts is $25 - x$ . Organize the information in a table:

Type	Number of Pounds	Price per pound ($)	Total Value ($)
Raisins	$x$	$4.50$	$4.50x$
Nuts	$25 - x$	$8$	$8(25 - x)$
Trail mix	$25$	$6.60$	$25(6.60) = 165$

Translate into an equation. The value of the raisins plus the value of the nuts equals the value of the trail mix: $4.50x + 8(25 - x) = 165$
Solve the equation:

Distribute $8$ : $4.50x + 200 - 8x = 165$
Combine like terms: $-3.50x + 200 = 165$
Subtract $200$ from both sides: $-3.50x = -35$
Divide both sides by $-3.50$ : $x = 10$ Find the number of pounds of nuts: $25 - 10 = 15$ .

Check: $4.50(10) + 8(15) \stackrel{?}{=} 25(6.60)$ $45 + 120 \stackrel{?}{=} 165$ $165 = 165$ $\checkmark$
Answer: Henning mixed $10$ pounds of raisins with $15$ pounds of nuts.

This example demonstrates that the mixture model ( $ext{amount} \cdot ext{value} = ext{total value}$ ) applies to products sold by weight. The known-total technique produces the expression $25 - x$ for the second ingredient, and solving the equation involves working with decimals.

Updated 2026-05-02

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