Question

an insect exhibit has 2 butterflies for every 3 caterpillars. there are 36 caterpillars in the exhibit.
(a) build a diagram that shows the number of butterflies and the number of caterpillars in the insect exhibit. make sure to do these steps.

• drag to adjust the number of sections in the diagram below.
• fill in the input box to show how many butterflies are in each section.

number of insects in each section:
diagram with total insects, butterflies, caterpillars labels and drag-to-build elements
(b) using the diagram, find the number of butterflies in the insect exhibit.
input box for butterflies and reset/cancel buttons

Explanation:

Step1: Find sections for caterpillars

The ratio of butterflies to caterpillars is \(2:3\). For caterpillars, each "section" of the ratio has 3 caterpillars. Total caterpillars are 36, so number of sections for caterpillars: \(36\div3 = 12\). So each section (for both butterflies and caterpillars) has 12 insects? Wait, no—wait, the ratio is 2 butterflies : 3 caterpillars. So the "3" in the ratio corresponds to caterpillars. So to find how many units (sections) of the ratio, we do \(36\div3 = 12\). So each "part" of the ratio (butterflies part: 2, caterpillars part: 3) is multiplied by 12? Wait, no, the number of insects in each section (the unit) for caterpillars: since 3 caterpillars per section, and total caterpillars 36, then number of sections for caterpillars is \(36\div3 = 12\)? Wait, no, maybe the "section" is the unit of the ratio. Wait, the problem says "Number of insects in each section". Wait, the ratio is 2 butterflies for every 3 caterpillars. So for caterpillars, 3 caterpillars per "section", and total caterpillars 36. So number of sections (groups) is \(36\div3 = 12\). Then, for butterflies, each section has 2 butterflies? Wait, no, the "Number of insects in each section"—wait, maybe the diagram has sections where each section for butterflies has 2 and caterpillars has 3, but scaled. Wait, no, let's re-express.

Wait, the ratio is \(2:3\) (butterflies : caterpillars). So let the number of butterflies be \(2x\), caterpillars \(3x\). Given \(3x = 36\), so \(x = 12\). Then butterflies are \(2x = 24\). But for part (a), "Number of insects in each section": since the ratio is 2:3, each "section" (the \(x\)) is 12? Wait, maybe the diagram is built with sections where each butterfly section has 2 and caterpillar section has 3, but scaled by \(x\). Wait, no, the problem says "Number of insects in each section"—maybe each section (for both butterflies and caterpillars) has \(x\) insects? Wait, no, the ratio is 2 butterflies to 3 caterpillars, so per "group" (section) of the ratio, there are 2 butterflies and 3 caterpillars. So total insects per group: 5, but the problem is about "number of insects in each section"—maybe the "section" is the \(x\) such that 3x = 36, so x = 12. So each section (the multiplier) is 12. Wait, maybe the "Number of insects in each section" is the value of \(x\), which is 12? Wait, no, let's check part (b). For part (b), number of butterflies: since ratio is 2:3, and \(3x = 36\), so \(x = 12\), then butterflies are \(2x = 24\). But for part (a), "Number of insects in each section"—maybe the section is the \(x\), so 12? Wait, let's do step by step.

Step1: Determine the scale factor (x)

Given ratio of butterflies (B) to caterpillars (C) is \(B:C = 2:3\). Let \(B = 2x\), \(C = 3x\). Given \(C = 36\), so \(3x = 36\). Solve for \(x\): \(x = 36\div3 = 12\).

Step2: Number of insects in each section

The "section" here is the \(x\), so each section (the unit of the ratio) has \(x = 12\) insects? Wait, no, the "Number of insects in each section"—maybe each section for butterflies has 2 and caterpillars has 3, but scaled by \(x\). Wait, no, the problem says "Number of insects in each section"—probably, each section (the group) has 12 insects? Wait, no, let's think again. For caterpillars, 3 insects per section (from the ratio 3), and total 36, so number of sections is \(36\div3 = 12\). So each section (group) has 3 caterpillars, and 2 butterflies. But the "Number of insects in each section"—maybe the number of insects per section (the count per section) for butterflies: 2, but no, the problem says "Number of insects in each section"—maybe it's the \(x\), which is 12? Wait, no, let's check part (b). For part (b), number of butterflies: \(2x = 2\times12 = 24\). So for part (a), the "Number of insects in each section" is 12? Wait, maybe the diagram has sections where each section for butterflies has 2 and caterpillars has 3, but when scaled, each section (the unit) is 12. Wait, maybe the "Number of insects in each section" is the value of \(x\), which is 12. So we fill 12 in the box.

Step2 (for part b): Calculate butterflies

Using the ratio, butterflies = \(2x\), and \(x = 12\), so \(2\times12 = 24\).

Answer (for part a): 12

Answer (for part b): 24

Answer:

(for part a): 12