Question

1. in a rice recipe, a ratio that relates the number of cups of water to the number of cups of rice is 2:5.

a. draw a tape diagram to represent this ratio.
b. how many cups of water should be used with 4 cups of rice?

1. the tape diagram shows that the ratio 4:3 is equivalent to the ratio 8:6.

a. draw a tape diagram to show that the ratio 4:3 is equivalent to the ratio 20:15.
b. yuna thinks that the ratio 4:3 is equivalent to the ratio 10:9 because 6 + 4 = 10 and 6 + 3 = 9. what is yunas mistake? what can she do to find an equivalent ratio?

Explanation:

1. a. Tape diagram for ratio \(2:5\) (water to rice)

• Explanation:
• A tape diagram for the ratio \(2:5\) (water to rice) would have two rectangles (or segments) for water and five rectangles (or segments) for rice. Each rectangle represents one unit. For example:
• Water: \(\square\square\)
• Rice: \(\square\square\square\square\square\)

1. b. Calculating cups of water for 4 cups of rice

• Step - by - Step Format:
• Explanation:
• Step 1: Set up the proportion
• Let \(x\) be the number of cups of water. The ratio of water to rice is \(2:5\), and we have \(x\) cups of water and 4 cups of rice. So the proportion is \(\frac{2}{5}=\frac{x}{4}\) (using the property of equivalent ratios, \(\text{water}_1:\text{rice}_1=\text{water}_2:\text{rice}_2\)).
• Step 2: Cross - multiply to solve for \(x\)
• Cross - multiplying gives us \(5x = 2\times4\). So \(5x=8\).
• Step 3: Solve for \(x\)
• Divide both sides of the equation \(5x = 8\) by 5. \(x=\frac{8}{5}=1.6\)

2. a. Tape diagram for ratio \(4:3\) and \(20:15\)

• Explanation:
• For the ratio \(4:3\), we have 4 rectangles (or segments) for the first quantity and 3 rectangles (or segments) for the second quantity.
• To show the ratio \(20:15\) (which is equivalent to \(4:3\) since \(\frac{20}{15}=\frac{4}{3}\)), we can group the rectangles.
• If each group of 5 rectangles (for the first quantity) and 5 rectangles (for the second quantity) (because \(20\div4 = 5\) and \(15\div3=5\)).
• First quantity (\(20\)): \(\underbrace{\square\square\square\square}_{\text{Group 1}}\underbrace{\square\square\square\square}_{\text{Group 2}}\underbrace{\square\square\square\square}_{\text{Group 3}}\underbrace{\square\square\square\square}_{\text{Group 4}}\underbrace{\square\square\square\square}_{\text{Group 5}}\)
• Second quantity (\(15\)): \(\underbrace{\square\square\square}_{\text{Group 1}}\underbrace{\square\square\square}_{\text{Group 2}}\underbrace{\square\square\square}_{\text{Group 3}}\underbrace{\square\square\square}_{\text{Group 4}}\underbrace{\square\square\square}_{\text{Group 5}}\)

2. b. Analyzing Yuna's mistake

• Answer - Explanation Format:
• Brief Explanations:
• Yuna's mistake is that she added values to get an equivalent ratio. Equivalent ratios are found by multiplying (or dividing) both terms of the ratio by the same non - zero number. For example, to find an equivalent ratio of \(4:3\), we can multiply both terms by \(k\) (where \(k

eq0\)). If \(k = 2\), we get \(8:6\); if \(k=5\), we get \(20:15\).

• Answer:
• Yuna's mistake is using addition to find an equivalent ratio. To find an equivalent ratio, she should multiply both terms of the ratio \(4:3\) by the same non - zero number.

Answer:

1. a. Tape diagram for ratio \(2:5\) (water to rice)

• Explanation:
• A tape diagram for the ratio \(2:5\) (water to rice) would have two rectangles (or segments) for water and five rectangles (or segments) for rice. Each rectangle represents one unit. For example:
• Water: \(\square\square\)
• Rice: \(\square\square\square\square\square\)

1. b. Calculating cups of water for 4 cups of rice

• Step - by - Step Format:
• Explanation:
• Step 1: Set up the proportion
• Let \(x\) be the number of cups of water. The ratio of water to rice is \(2:5\), and we have \(x\) cups of water and 4 cups of rice. So the proportion is \(\frac{2}{5}=\frac{x}{4}\) (using the property of equivalent ratios, \(\text{water}_1:\text{rice}_1=\text{water}_2:\text{rice}_2\)).
• Step 2: Cross - multiply to solve for \(x\)
• Cross - multiplying gives us \(5x = 2\times4\). So \(5x=8\).
• Step 3: Solve for \(x\)
• Divide both sides of the equation \(5x = 8\) by 5. \(x=\frac{8}{5}=1.6\)

2. a. Tape diagram for ratio \(4:3\) and \(20:15\)

• Explanation:
• For the ratio \(4:3\), we have 4 rectangles (or segments) for the first quantity and 3 rectangles (or segments) for the second quantity.
• To show the ratio \(20:15\) (which is equivalent to \(4:3\) since \(\frac{20}{15}=\frac{4}{3}\)), we can group the rectangles.
• If each group of 5 rectangles (for the first quantity) and 5 rectangles (for the second quantity) (because \(20\div4 = 5\) and \(15\div3=5\)).
• First quantity (\(20\)): \(\underbrace{\square\square\square\square}_{\text{Group 1}}\underbrace{\square\square\square\square}_{\text{Group 2}}\underbrace{\square\square\square\square}_{\text{Group 3}}\underbrace{\square\square\square\square}_{\text{Group 4}}\underbrace{\square\square\square\square}_{\text{Group 5}}\)
• Second quantity (\(15\)): \(\underbrace{\square\square\square}_{\text{Group 1}}\underbrace{\square\square\square}_{\text{Group 2}}\underbrace{\square\square\square}_{\text{Group 3}}\underbrace{\square\square\square}_{\text{Group 4}}\underbrace{\square\square\square}_{\text{Group 5}}\)

2. b. Analyzing Yuna's mistake

• Answer - Explanation Format:
• Brief Explanations:
• Yuna's mistake is that she added values to get an equivalent ratio. Equivalent ratios are found by multiplying (or dividing) both terms of the ratio by the same non - zero number. For example, to find an equivalent ratio of \(4:3\), we can multiply both terms by \(k\) (where \(k

eq0\)). If \(k = 2\), we get \(8:6\); if \(k=5\), we get \(20:15\).

• Answer:
• Yuna's mistake is using addition to find an equivalent ratio. To find an equivalent ratio, she should multiply both terms of the ratio \(4:3\) by the same non - zero number.