Question

23 ar3.3 display equivalent part-to-part and part-to-part-to-whole ratios; by completing a two-column or three-column table with missing values (l3) a paint mixture uses ¾ gallon of blue paint for every ½ gallon of white paint. complete the table. blue paint white paint total paint ¾ ½ 1 ½ 2 ½ 3 a smoothie recipe uses 2 cups of yogurt for every 2¼ cups of fruit. complete the table. yogurt fruit total cups 2 2¼ 1 9

Explanation:

Step1: Solve Paint Mixture Table (Row 1: Total)

To find total paint, add blue and white paint.
Blue: $\frac{3}{4}$, White: $\frac{1}{2}=\frac{2}{4}$.
Total: $\frac{3}{4}+\frac{2}{4}=\frac{5}{4}=1\frac{1}{4}$.

Step2: Solve Paint Mixture Table (Row 2: White Paint)

Total paint: $2\frac{1}{2}=\frac{5}{2}$, Blue: $1\frac{1}{2}=\frac{3}{2}$.
White = Total - Blue: $\frac{5}{2}-\frac{3}{2}=\frac{2}{2}=1$.

Step3: Solve Paint Mixture Table (Row 3: Blue Paint)

Ratio of Blue to White: $\frac{3}{4}:\frac{1}{2}=3:2$ (simplify by multiplying by 4: $3:2$).
If White = 3, let Blue = $x$. Then $\frac{x}{3}=\frac{3}{2}\implies x=\frac{9}{2}=4\frac{1}{2}$.
Total = Blue + White = $4\frac{1}{2}+3=7\frac{1}{2}$.

Step4: Solve Smoothie Table (Row 1: Total)

Add yogurt and fruit: $2 + 2\frac{1}{4}=2+\frac{9}{4}=\frac{8}{4}+\frac{9}{4}=\frac{17}{4}=4\frac{1}{4}$.

Step5: Solve Smoothie Table (Row 2: Fruit)

Ratio of Yogurt to Fruit: $2:2\frac{1}{4}=2:\frac{9}{4}=8:9$ (multiply by 4).
If Yogurt = 1, let Fruit = $y$. $\frac{1}{y}=\frac{8}{9}\implies y=\frac{9}{8}=1\frac{1}{8}$.
Total = $1 + 1\frac{1}{8}=2\frac{1}{8}$.

Step6: Solve Smoothie Table (Row 3: Yogurt)

Ratio 8:9 (Yogurt:Fruit). If Fruit = 9, Yogurt = 8 (from ratio 8:9).
Total = $8 + 9=17$.

Paint Mixture Table:

Blue Paint	White Paint	Total Paint
$1\frac{1}{2}$	$1$	$2\frac{1}{2}$
$4\frac{1}{2}$	$3$	$7\frac{1}{2}$

Smoothie Table:

Yogurt	Fruit	Total cups
$1$	$1\frac{1}{8}$	$2\frac{1}{8}$
$8$	$9$	$17$

Answer:

• Paint Mixture (Row 1 Total): $\boldsymbol{1\frac{1}{4}}$; (Row 2 White): $\boldsymbol{1}$; (Row 3 Blue): $\boldsymbol{4\frac{1}{2}}$, (Row 3 Total): $\boldsymbol{7\frac{1}{2}}$.
• Smoothie (Row 1 Total): $\boldsymbol{4\frac{1}{4}}$; (Row 2 Fruit): $\boldsymbol{1\frac{1}{8}}$, (Row 2 Total): $\boldsymbol{2\frac{1}{8}}$; (Row 3 Yogurt): $\boldsymbol{8}$, (Row 3 Total): $\boldsymbol{17}$.