A supplier sells $2 \frac{1}{4}$ pounds of mulch for every $1 \frac{1}{3}$ pounds of gravel. The supplier sells $172$ pounds of mulch and gravel combined. How many pounds of each item does the supplier sell? $\newline$ The supplier sells $\square$ pounds of mulch and $\square$ pounds of gravel.

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Write a ratio: word problems

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Q. A supplier sells

2 \frac{1}{4}

pounds of mulch for every

1 \frac{1}{3}

pounds of gravel. The supplier sells

172

pounds of mulch and gravel combined. How many pounds of each item does the supplier sell?

\newline

The supplier sells

\square

pounds of mulch and

\square

pounds of gravel.

Convert to Improper Fractions: Convert the mixed fractions to improper fractions to simplify calculations. $\newline$ $2\left(\frac{1}{4}\right)$ pounds of mulch is equivalent to $\left(2\times 4 + 1\right)/4 = \frac{9}{4}$ pounds of mulch. $\newline$ $1\left(\frac{1}{3}\right)$ pounds of gravel is equivalent to $\left(1\times 3 + 1\right)/3 = \frac{4}{3}$ pounds of gravel.
Set Up Ratio: Set up the ratio of mulch to gravel using the improper fractions. $\newline$ The ratio of mulch to gravel is $\frac{9}{4}$ to $\frac{4}{3}$ .
Find Common Denominator: Find the common denominator to combine the ratio into a single fraction that represents the total weight of mulch and gravel sold per unit. $\newline$ The common denominator for $4$ and $3$ is $12$ . Multiply the numerators by the necessary factors to get the equivalent fractions with a denominator of $12$ . $\newline$ For mulch: $(\frac{9}{4}) \times (\frac{3}{3}) = \frac{27}{12}$ $\newline$ For gravel: $(\frac{4}{3}) \times (\frac{4}{4}) = \frac{16}{12}$
Add Fractions: Add the fractions representing mulch and gravel to find the total weight per unit. $\newline$ Total weight per unit = $(\frac{27}{12}) + (\frac{16}{12}) = \frac{43}{12}$
Calculate Number of Units: Calculate the number of units sold by dividing the total weight sold by the weight per unit. $\newline$ Total weight sold = $172$ pounds $\newline$ Weight per unit = $\frac{43}{12}$ pounds $\newline$ Number of units = Total weight sold / Weight per unit = $\frac{172}{(\frac{43}{12})}$
Perform Division: Perform the division to find the number of units. $\newline$ Number of units = $172 \times (12/43) = (172 \times 12) / 43 = 2064 / 43 = 48$
Calculate Mulch Sold: Calculate the amount of mulch sold by multiplying the number of units by the weight of mulch per unit. $\newline$ Amount of mulch = Number of units $\times$ Weight of mulch per unit = $48 \times \left(\frac{9}{4}\right)$
Perform Multiplication for Mulch: Perform the multiplication to find the amount of mulch sold. $\newline$ Amount of mulch = $48 \times \left(\frac{9}{4}\right) = \left(48 \times 9\right) / 4 = \frac{432}{4} = 108$ pounds
Calculate Gravel Sold: Calculate the amount of gravel sold by multiplying the number of units by the weight of gravel per unit. $\newline$ Amount of gravel $= \text{Number of units} \times \text{Weight of gravel per unit} = 48 \times \left(\frac{4}{3}\right)$
Perform Multiplication for Gravel: Perform the multiplication to find the amount of gravel sold. $\newline$ Amount of gravel = $48 \times \left(\frac{4}{3}\right) = \left(48 \times 4\right) / 3 = \frac{192}{3} = 64$ pounds