<p>Last week, the Starts Bakery made 3 kinds of cakes. One half of the cakes were made with 4 eggs each, 2/3 of the rest of the cakes were made with 3 eggs each, and the remaining 54 cakes were made with 2 eggs each. What is the total amount of eggs used to make all these cakes? </p>
<p>A clear and concise solution would be appreciated.</p>
<p>1080 eggs… am i right?</p>
<p>I’m also getting 1080.</p>
<p>Let’s say x is the total number of cakes.</p>
<p>1-((x/2)+((2/3)(x/2)))=54.</p>
<p>x=324 cakes.</p>
<p>x/2=162. Number of eggs for these cakes is 162*4=648 eggs.</p>
<p>(2/3)(162)=108. Number of eggs for these cakes is 108*3=324 eggs.</p>
<p>For the remaining 54 cakes, number of eggs is 54*2=108 eggs.</p>
<p>Total eggs=648+324+108=1080 eggs</p>
<p>what he said ^^ haha</p>
<p>314159265</p>
<p>I lost you at 1-((x/2)+((2/3)(x/2)))=54.
How did you get that again? And when I try to find x I get like -63.5?</p>
<p>Last week, the Starts Bakery made 3 kinds of cakes. One half of the cakes were made with 4 eggs each, 2/3 of the rest of the cakes were made with 3 eggs each, and the remaining 54 cakes were made with 2 eggs each. What is the total amount of eggs used to make all these cakes?</p>
<p>54 cakes = 1 - 0.5 * 2/3 + 0.5 = 1 - 0.5 * 5/3 = 1 - 2.5 / 3 = 1 - 5 / 6 = 1 / 6
Total amount of cakes = 54 / (1/6) = 54 * 6 = 324</p>
<p>“One half of the cakes were made with 4 eggs each”:
0.5 * 324 * 4 = 648
“2/3 of the rest of the cakes were made with 3 eggs each”:
1 / 3 * 0.5 * 324 * 3 = 0.5 * 324 = 162
“, and the remaining 54 cakes were made with 2 eggs each”:
54 * 2 = 108</p>
<p>648 + 324 + 108 = 1080</p>
<p>If you are going to use algebra,</p>
<p>x/2 - (2/3)(x/2) = 54</p>
<p>But this is not necessary. Think about the numbers before you waste time:</p>
<p>After you take away half, and then 2/3 of what’s left, you have 54 cakes. So those 54 cakes are 1/3 of what’s left (after half were taken away). So 54 times 3 = 162 which must be the half that were left. And 162 times 2 = 324, the cakes you started with.</p>
<p>Of course this can go much faster than the way described above (by me), but just start slow. When you do a couple of these exercises, you’ll probably only have to do 1-2 ‘real’ calculations. </p>
<p>54 = 1/6. Then you need for the first one 54 * 12, the second one 54 * 6 and the third one 54 * 2, which is 54 * 20 = 1080. No need for a calculator.</p>
<p>i also got 1080…</p>
<p>What number question was this in the grid ins? the last one?</p>
<p>I think it was the 3rd to last one. (Number 15 I think)</p>
<p>Get rid of the big numbers and start with 6 cakes:</p>
<p>A. 3 x 4 eggs = 12
B. 2 x 3 eggs = 6</p>
<h1>C. 1 x 2 eggs = 2 </h1>
<p>Total … 20</p>
<p>Since C represents 1/10 of total and uses 108 eggs, the answer is 1080. Isn’t nice when you can multiply by 10? :)</p>
<p>PS Please note that the total number of eggs used in A (648) and B (324) is NOT relevant. If you used those numbers, you wasted time. Only C and total are needed.</p>
<p>Xiggi, I don’t think that’s the most efficient way to do this. But if you do want to use this method, you might be better off not multiplying C by 2 and then by 10, but immediately by 20. Saves you one calculation.</p>
<h1>1. (1/2)*(2/3)= 1/3</h1>
<h1>2. 1/3 + 1/2= 5/6</h1>
<h1>3. 1-5/6=1/6</h1>
<h1>4. 54/(1/6)=324=Total Number Of Eggs</h1>
<h1>5. First Batch of Cakes were made by (324/2 * 4 Eggs)= 648</h1>
<pre><code> Second Batch of Cakes were made by (1/22/33243)=324
Third and Final Batch of Cakes were made by (542)= 108
</code></pre>
<h1>6 Add All 3 up 648+324+108= 1080 Final Answer</h1>
<p>lol I still got the SAT skills</p>
<p>
</p>
<p>Dutchguy, you lost me there! How could multiply by a number (20) that you do not know yet. You need the 2 before you can get the 20. I think you missed the point. </p>
<p>Unless you meant to multiply the 54 by 20, which is exactly the same as 54.2.10 and represents one MENTAL operation and not two calculations. </p>
<p>As far as being efficient, I am not sure how you could make what I suggested more efficient as it took all but 8 to 12 seconds to write down the answer. Remember that the explanation given is meant to help people follow; it’s not the way one solves the problem during the test because one does not need to write the steps as we do here. </p>
<p>All that my paper shows is this:</p>
<p>3> 12
2> 6
1> 2 … 1
==…== >>> 10.54.2<br>
6> 20…10</p>
<p>PS Your quick solution entails this
54 cakes = 1 - 0.5 * 2/3 + 0.5 = 1 - 0.5 * 5/3 = 1 - 2.5 / 3 = 1 - 5 / 6 = 1 / 6
54 = 1/6. Then you need for the first one 54 * 12, the second one 54 * 6 and the third one 54 * 2, which is 54 * 20 = 1080. </p>
<p>I simply skipped your first line by going to six cakes immediately (because of the 3<em>2</em>1) … if there is a line that can be skipped for efficiency is the algebraic manipulation in your first line. :)</p>
<p>
</p>
<p>Sorry I made this look so complicated. I’m an algebra person, so you might like Xiggi’s solution better. :P</p>
<p>But the equation above basically says this: We know that there are cakes made with 4 eggs each, cakes made with 3 eggs each, and cakes made with 2 eggs each. Each group (separated by number of eggs needed) contributes to the total number of cakes. The variable x in my equation is the total number of cakes. Since we know that half the cakes require four eggs, that’s 50%. That’s also x/2 (half of the total number of cakes). </p>
<p>Since we’ve taken away x/2 already (those are the cakes that need 4 eggs), the cakes that need 2 eggs plus those that need 3 eggs make up the other 50% or x/2. We also know that (2/3) of these cakes require three eggs. That can be shown as (2/3) times (x/2).</p>
<p>We know that what’s left after we take away the x/2 [cakes that require 4 eggs] and the (x/2)(2/3) [cakes that require 3 eggs], we’re left with 54 cakes.</p>
<p>Well, when you add up the (x/2) and the (2/3)(x/2) <a href=“which%20I’ll%20now%20write%20as%20(x/3)%20since%20I%20multiplied%20it”>b</a>**, that’s almost 100%. But what’s missing from that is the 54. So what we do is set up the equation of 100%-x/2-x/3=54. We change the 100% simply into 1 and solve. We see that x/2 and x/3 form (5/6)x, so we see that 54 must equal (x/6). x=total number of cakes=54*6=324.</p>
<p>If you understand Xiggi’s solution more than mine, learn to use Xiggi’s for sure. But hopefully this cleared up some of the confusion!</p>
<p>/sigh</p>
<p>You are making it way too complicated. ETS spends a fortune building traps. Anyone who used numbers such as 648, 324, or 162 fell into ETS sinking time trap. </p>
<p>I also love to find a solution that involves a neat algebraic equation. However, an equation that uses the wrong variable is a poor use of algebra. So, let me repeat two points:</p>
<ol>
<li>The total number of cakes is NOT relevant<br></li>
<li>The number of cakes or eggs used in the first two batches is NOT relevant. </li>
</ol>
<p>If you computed any of those things, you WASTED your time. If you want to use algebra, you need an equation that addresses the question, namely number of EGGS, not cakes! </p>
<p>The only thing that is relevant is the ratio of how many eggs of the third type of cake to the total number of eggs. </p>
<p>The SAT is a reasoning test that happens to use VERY basic math. The key to do well is to keep things as simple as possible.</p>
<p>
</p>
<p>I wouldn’t think it’s that serious. Yes, your method probably took you less time than my method took me. I would agree that attempting to generate the equations that I used would be a trap if the student weren’t well-versed in converting word problems into algebraic expressions. However, once the number of cakes is found, only simple multiplication and addition is left. It took me about 90 seconds to do this entire question by hand with my method.</p>
<p>With that being said, I believe that the best method is the method that the student actually understands how to do properly and quickly. Xiggi, while your solution makes sense, it would have taken me much more time to figure out the question your way than it did for me to solve using algebra. However, that may not be the case with other kids (and definitely not the case with you).</p>
<p>Almost in xiggi’s footsteps.</p>
<p>It’s helpful to remember that
1/2+1/3+1/6=1 and
1/2:1/3:1/6=3:2:1 (and =180:120:60, nice to use on a pie chart).
You can tell, I like ratios. :)</p>
<p>Ratio of cakes = 3:2:1
Ratio of eggs = (3x4) : (2x3) : (1x2) = 12:6:2 = 6:3:1
1 part corresponds to (54x2) eggs, total parts = 6+3+1 = 10, therefore
total eggs = 108x10 = 1080.</p>
<p>^^ What can I say? The OP asked for “a clear and concise solution”<br>
I followed the narrative and looked at the elements that were important.</p>
<p>Last week, the Starts Bakery made 3 kinds of cakes. One half of the cakes were made with 4 eggs each, 2/3 of the rest of the cakes were made with 3 eggs each, and the remaining 54 cakes were made with 2 eggs each. What is the total amount of **eggs **used to make all these cakes? </p>
<p>What happens when you see 1/2, (2/3X1/2), and “the rest.” Isn’t it natural to turn that into a simple equation? 3/6 + 2/6 + 1/6 = 1. That is why picked 6 to compute my ratio. </p>
<p>From there it was a piece of cake. :)</p>
<p>PS Haha, crossposted with GCF! I like ratios too. I also like funny things such as 1+2+3 = 1<em>2</em>3. Try to find three other numbers that can do that.</p>
<p>
</p>
<p>Okay xiggi, I stand corrected. I just got my calculation and yours mixed up. What I do recommend is that you just leave 54*2 sit there and then, when you’ve got to multiply by 10, you immediately multiply by 20. </p>
<p>
</p>
<p>Well, yes, you’re right, that does look like cumbersome. But actually I am doing the same thing but without mental calculation. When you just do it mental, you rapidly (just as with your method) get to 1/6. Then you can go on to the second line.</p>
<p>But well, I like my way and, I believe, more like xiggi’s way. Both ways will get you there in time. :)</p>