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Sagot :
To determine how many units of Product Y Arnott's will sell, we need to follow a series of steps.
### Step 1: Calculate the Contribution Margin for Each Product
The contribution margin is determined by subtracting the variable cost per unit from the sales price per unit for each product.
For Product X:
[tex]\[ \text{Contribution Margin} (cm_X) = \text{Sales Price} - \text{Variable Cost} = 9 - 4 = 5 \][/tex]
For Product Y:
[tex]\[ \text{Contribution Margin} (cm_Y) = \text{Sales Price} - \text{Variable Cost} = 14 - 8 = 6 \][/tex]
For Product Z:
[tex]\[ \text{Contribution Margin} (cm_Z) = \text{Sales Price} - \text{Variable Cost} = 15 - 7 = 8 \][/tex]
### Step 2: Determine the Total Sales Mix
The total sales mix is the sum of the individual sales mix units for products X, Y, and Z.
[tex]\[ \text{Total Sales Mix} = 4 + 4 + 2 = 10 \][/tex]
### Step 3: Calculate the Weighted Average Contribution Margin
The weighted average contribution margin is obtained by multiplying each product's contribution margin by its proportion of the total sales mix and then summing these values.
For Product X:
[tex]\[ \frac{\text{Sales Mix of X}}{\text{Total Sales Mix}} \times cm_X = \frac{4}{10} \times 5 = 2 \][/tex]
For Product Y:
[tex]\[ \frac{\text{Sales Mix of Y}}{\text{Total Sales Mix}} \times cm_Y = \frac{4}{10} \times 6 = 2.4 \][/tex]
For Product Z:
[tex]\[ \frac{\text{Sales Mix of Z}}{\text{Total Sales Mix}} \times cm_Z = \frac{2}{10} \times 8 = 1.6 \][/tex]
Summing these contributions gives us the weighted average contribution margin:
[tex]\[ \text{Weighted CM} = 2 + 2.4 + 1.6 = 6.0 \][/tex]
### Step 4: Calculate the Break-Even Point in Total Units
The break-even point in units is determined by dividing the total fixed costs by the weighted average contribution margin.
[tex]\[ \text{Break-Even Units} = \frac{\text{Fixed Costs}}{\text{Weighted CM}} = \frac{144000}{6.0} = 24000 \text{ units} \][/tex]
### Step 5: Determine the Number of Units of Product Y Sold
The number of units of Product Y sold is calculated by multiplying the break-even units by the proportion of Product Y in the sales mix.
[tex]\[ \text{Units of Y Sold} = \text{Break-Even Units} \times \frac{\text{Sales Mix of Y}}{\text{Total Sales Mix}} = 24000 \times \frac{4}{10} = 9600 \text{ units} \][/tex]
Therefore, Arnott's will sell 9,600 units of Product Y at the break-even point.
### Step 1: Calculate the Contribution Margin for Each Product
The contribution margin is determined by subtracting the variable cost per unit from the sales price per unit for each product.
For Product X:
[tex]\[ \text{Contribution Margin} (cm_X) = \text{Sales Price} - \text{Variable Cost} = 9 - 4 = 5 \][/tex]
For Product Y:
[tex]\[ \text{Contribution Margin} (cm_Y) = \text{Sales Price} - \text{Variable Cost} = 14 - 8 = 6 \][/tex]
For Product Z:
[tex]\[ \text{Contribution Margin} (cm_Z) = \text{Sales Price} - \text{Variable Cost} = 15 - 7 = 8 \][/tex]
### Step 2: Determine the Total Sales Mix
The total sales mix is the sum of the individual sales mix units for products X, Y, and Z.
[tex]\[ \text{Total Sales Mix} = 4 + 4 + 2 = 10 \][/tex]
### Step 3: Calculate the Weighted Average Contribution Margin
The weighted average contribution margin is obtained by multiplying each product's contribution margin by its proportion of the total sales mix and then summing these values.
For Product X:
[tex]\[ \frac{\text{Sales Mix of X}}{\text{Total Sales Mix}} \times cm_X = \frac{4}{10} \times 5 = 2 \][/tex]
For Product Y:
[tex]\[ \frac{\text{Sales Mix of Y}}{\text{Total Sales Mix}} \times cm_Y = \frac{4}{10} \times 6 = 2.4 \][/tex]
For Product Z:
[tex]\[ \frac{\text{Sales Mix of Z}}{\text{Total Sales Mix}} \times cm_Z = \frac{2}{10} \times 8 = 1.6 \][/tex]
Summing these contributions gives us the weighted average contribution margin:
[tex]\[ \text{Weighted CM} = 2 + 2.4 + 1.6 = 6.0 \][/tex]
### Step 4: Calculate the Break-Even Point in Total Units
The break-even point in units is determined by dividing the total fixed costs by the weighted average contribution margin.
[tex]\[ \text{Break-Even Units} = \frac{\text{Fixed Costs}}{\text{Weighted CM}} = \frac{144000}{6.0} = 24000 \text{ units} \][/tex]
### Step 5: Determine the Number of Units of Product Y Sold
The number of units of Product Y sold is calculated by multiplying the break-even units by the proportion of Product Y in the sales mix.
[tex]\[ \text{Units of Y Sold} = \text{Break-Even Units} \times \frac{\text{Sales Mix of Y}}{\text{Total Sales Mix}} = 24000 \times \frac{4}{10} = 9600 \text{ units} \][/tex]
Therefore, Arnott's will sell 9,600 units of Product Y at the break-even point.
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