Question

sara and geena can both do flips in the air. the ratio of the number of flips sara can do to the number of flips geena can do is 3:8. geena can do 120 more flips than sara. if sara increases the number of flips she can do by 3 and geena decreases the number of flips she can do by 12, what will be the new ratio of the number of flips sara can do to the number of flips geena can do?

Explanation:

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<reasoning>

Define variables using ratios

Using the Ratios knowledge point
\[

$$\begin{aligned} &\text{Let } S = 3x\\ &\text{Let } G = 8x \end{aligned}$$

\]

Set up the equation

Using the Solving Algebraic Equations knowledge point
\[

$$\begin{aligned} &G = S + 120\\ &8x = 3x + 120 \end{aligned}$$

\]

Solve for the multiplier

Using the Solving Algebraic Equations knowledge point
\[

$$\begin{aligned} &5x = 120\\ &x = 24 \end{aligned}$$

\]

Calculate the original and new values

Using the Ratios knowledge point
\[

$$\begin{aligned} &S = 3 \times 24 = 72\\ &G = 8 \times 24 = 192\\ &S_{\text{new}} = 72 + 3 = 75\\ &G_{\text{new}} = 192 - 12 = 180 \end{aligned}$$

\]

Simplify the new ratio

Using the Ratios knowledge point
\[

$$\begin{aligned} &\frac{S_{\text{new}}}{G_{\text{new}}} = \frac{75}{180} = \frac{5}{12}\\ &\text{Ratio} = 5:12 \end{aligned}$$

\]
</reasoning>

<answer>
Sara and Geena can both do flips in the air. The ratio of the number of flips Sara can do to the number of flips Geena can do is 3:8. Geena can do 120 more flips than Sara. If Sara increases the number of flips she can do by 3 and Geena decreases the number of flips she can do by 12, what will be the new ratio of the number of flips Sara can do to the number of flips Geena can do? <blank>\(5:12\)</blank>
</answer>

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"question_type": "Fill-in-the-blank",
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"Mathematics",
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"Ratio Word Problems"
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Answer:

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<reasoning>

Define variables using ratios

Using the Ratios knowledge point
\[

$$\begin{aligned} &\text{Let } S = 3x\\ &\text{Let } G = 8x \end{aligned}$$

\]

Set up the equation

Using the Solving Algebraic Equations knowledge point
\[

$$\begin{aligned} &G = S + 120\\ &8x = 3x + 120 \end{aligned}$$

\]

Solve for the multiplier

Using the Solving Algebraic Equations knowledge point
\[

$$\begin{aligned} &5x = 120\\ &x = 24 \end{aligned}$$

\]

Calculate the original and new values

Using the Ratios knowledge point
\[

$$\begin{aligned} &S = 3 \times 24 = 72\\ &G = 8 \times 24 = 192\\ &S_{\text{new}} = 72 + 3 = 75\\ &G_{\text{new}} = 192 - 12 = 180 \end{aligned}$$

\]

Simplify the new ratio

Using the Ratios knowledge point
\[

$$\begin{aligned} &\frac{S_{\text{new}}}{G_{\text{new}}} = \frac{75}{180} = \frac{5}{12}\\ &\text{Ratio} = 5:12 \end{aligned}$$

\]
</reasoning>

<answer>
Sara and Geena can both do flips in the air. The ratio of the number of flips Sara can do to the number of flips Geena can do is 3:8. Geena can do 120 more flips than Sara. If Sara increases the number of flips she can do by 3 and Geena decreases the number of flips she can do by 12, what will be the new ratio of the number of flips Sara can do to the number of flips Geena can do? <blank>\(5:12\)</blank>
</answer>

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