Question

multiple choice 1 point as of 2009, there were about 225 quarter for every 2 half - dollar coins in circulation. if there were 380 million quarters in circulation that year, approximately how many total quarters and fifty - cent coins, in millions were there in circulation? 415 493 383 605

Explanation:

Step1: Determine the ratio of quarters to half - dollar coins

The ratio of quarters to half - dollar (fifty - cent) coins is 225 quarters for every 2 half - dollar coins. So the ratio of quarters (\(Q\)) to half - dollar coins (\(H\)) is \(\frac{Q}{H}=\frac{225}{2}\), which can be rewritten as \(H=\frac{2}{225}Q\).

Step2: Substitute the number of quarters

We know that the number of quarters \(Q = 380\) million. Substitute \(Q = 380\) into the formula for \(H\): \(H=\frac{2}{225}\times380\). First, calculate \(\frac{380}{225}\approx1.6889\), then multiply by 2: \(1.6889\times2 = 3.3778\)? Wait, no, wait. Wait, the ratio is 225 quarters per 2 half - dollar coins. So if we have \(Q\) quarters, the number of half - dollar coins \(H=\frac{2}{225}Q\)? Wait, no, let's re - express the ratio. Let the number of half - dollar coins be \(x\). The ratio of quarters to half - dollar coins is \(225:2\), so \(\frac{380}{x}=\frac{225}{2}\). Cross - multiply: \(225x=380\times2 = 760\). Then \(x=\frac{760}{225}\approx3.3778\)? No, that can't be right. Wait, maybe I got the ratio reversed. The problem says "225 quarter for every 2 half - dollar coins", so quarters : half - dollar = 225:2. So total coins = quarters+half - dollar. So total coins \(T=Q + H\), and \(H=\frac{2}{225}Q\). Wait, no, if 225 quarters correspond to 2 half - dollar coins, then for \(Q = 380\) quarters, the number of half - dollar coins \(H=\frac{2}{225}\times380\)? Wait, no, let's do it again. Let's set up the proportion: \(\frac{225}{2}=\frac{380}{H}\), where \(H\) is the number of half - dollar coins. Cross - multiplying gives \(225H=380\times2 = 760\), so \(H=\frac{760}{225}\approx3.3778\)? That seems wrong. Wait, maybe the ratio is 225 quarters to 2 half - dollar coins, so the total number of coins of both types is \(225 + 2=227\) for a group? No, the problem says "how many total quarters and fifty - cent coins". So total coins \(T=Q + H\). We know \(Q = 380\) million, and \(\frac{Q}{H}=\frac{225}{2}\), so \(H=\frac{2Q}{225}\). Then \(T=Q+\frac{2Q}{225}=Q(1 + \frac{2}{225})=Q\times\frac{225 + 2}{225}=Q\times\frac{227}{225}\). Now substitute \(Q = 380\) million: \(T = 380\times\frac{227}{225}\). Calculate \(\frac{227}{225}\approx1.00889\), then \(380\times1.00889\approx380 + 380\times0.00889\approx380+3.38\approx383.38\) million. Which is approximately 383 million.

Answer:

383 (the option with 383)