Question

prisms
pyramids
the table shows the number of prisms and the number of total blocks that cora will use on each of her towers.
based on the ratio, complete the missing values in the table.
tower prisms total blocks
a 10
d 35

Explanation:

Step1: Determine the ratio of prisms to total blocks

From the diagram, we see that there are 4 prisms and 1 pyramid, so total blocks for that arrangement is \(4 + 1=5\). So the ratio of prisms to total blocks is \(4:5\) (or \(\frac{4}{5}\) when considering prisms over total blocks, but we can also think of total blocks as \(\frac{5}{4}\) times the number of prisms since for 4 prisms, total blocks are 5). Wait, actually, let's check: prisms = 4, total blocks = prisms + pyramids = 4 + 1 = 5. So the ratio of prisms to total blocks is \(4:5\), meaning total blocks = \(\frac{5}{4}\times\) number of prisms.

Step2: Calculate total blocks for Tower A

For Tower A, prisms = 16. Using the ratio, total blocks = \(\frac{5}{4}\times16\). Calculate \(\frac{5}{4}\times16 = 5\times4 = 20\)? Wait, no, wait: wait, prisms are 4, total blocks 5. So the ratio of prisms to total is 4:5, so total blocks = (5/4)prisms. Wait, 4 prisms give 5 total, so 1 prism gives 5/4 total? No, that can't be. Wait, maybe the ratio is prisms : total blocks = 4 : 5, so if prisms are \(p\), total blocks \(t\), then \(t=\frac{5}{4}p\). Wait, for \(p = 4\), \(t = 5\), which works. So for \(p = 16\), \(t=\frac{5}{4}\times16 = 20\)? Wait, no, that seems low. Wait, maybe I got the ratio reversed. Wait, prisms are 4, pyramids are 1, so total blocks are 4 + 1 = 5. So the number of prisms is 4 when total blocks are 5. So the ratio of prisms to total blocks is 4:5, so prisms = (4/5)total blocks, or total blocks = (5/4)prisms. Wait, let's check with the diagram: 4 prisms and 1 pyramid, total 5 blocks. So yes, for every 4 prisms, there is 1 pyramid, so total blocks = prisms + (prisms / 4). Because pyramids are 1 when prisms are 4, so pyramids = prisms / 4. So total blocks = prisms + (prisms / 4) = (5/4)prisms. So for prisms = 16, total blocks = (5/4)16 = 20. Wait, but let's check the other way: for total blocks, if we know total blocks, prisms = (4/5)total blocks. Let's do Tower B: total blocks = 35. So prisms = (4/5)35 = 28. Let's verify: if prisms = 28, pyramids = 28 / 4 = 7, total blocks = 28 + 7 = 35. Yes, that works. So for Tower A: prisms = 16, pyramids = 16 / 4 = 4, total blocks = 16 + 4 = 20. For Tower B: total blocks = 35, pyramids = 35 - prisms, and pyramids = prisms / 4, so 35 - prisms = prisms / 4 → 35 = prisms + prisms / 4 → 35 = (5/4)prisms → prisms = 35*(4/5) = 28.

Step3: Calculate prisms for Tower B

For Tower B, total blocks = 35. We know that pyramids = prisms / 4 (since for 4 prisms, 1 pyramid). So total blocks = prisms + pyramids = prisms + (prisms / 4) = (5/4)prisms. So solving for prisms: prisms = (4/5)total blocks = (4/5)35 = 28.

Wait, but let's re-express:

From the diagram, number of prisms (P) and pyramids (Py) have a ratio P:Py = 4:1 (since 4 prisms and 1 pyramid). So Py = P / 4. Total blocks (T) = P + Py = P + P/4 = (5/4)P. So:

For Tower A: P = 16, so T = (5/4)*16 = 20.

For Tower B: T = 35, so 35 = (5/4)P → P = 35*(4/5) = 28.

Answer:

For Tower A, total blocks = 20; for Tower B, prisms = 28.

(Assuming the table is:

Tower | Prisms | Total blocks

A | 16 | 20

B | 28 | 35)