last week's MGRE Math Beast Challenge: correct!

Here's what MGRE has to say about last week's Math Beast Challenge problem:

Let’s start by translating this into algebra.

Abe has k ketchup packets, Beata has m mustard packets, Cruz has s soy sauce packets, and Dion has b barbecue sauce packets. We are trying to find the smallest possible value for k + m + s + b.

We know that 2k = 9m = 7s = 15b. In other words: 2k, 9m, 7s and 15b each equal the same integer number, but what is that number? It would have to be cleanly divisible by 2, 9, 7, and 15. In order to minimize the number of packets owned by the group as a whole, we would want to find the smallest such number. That would be least common multiple of 2, 9, 7, and 15.

First, we find the factors of each number:
2 = 2
9 = 3 * 3
7 = 7
15 = 3 * 5

Then we multiply only the necessary factors together:
2 * 3 * 3 * 7 * 5 = 630.

Note that we leave out a 3, compared to the list of all factors above. Remember that we only include the factors needed to build each of the starting numbers individually. With two 3’s and a 5, we could make either 9 or 15. That’s good enough for a least common multiple.

Now we know that 2k, 9m, 7s, and 15b each equals 630. Let’s find how many packets each person has.

2k = 630, so k = 315 9m = 630, so m = 70 7s = 630, so s = 90 15b = 630, so b = 42

To finish, just add together the individual number of packets (k + m + s + b) = 315 + 70 + 90 + 42 = 517.

It is worth analyzing how the other choices all represent a possible mistake, a quality that makes this question harder than it would be with different answer choices:

(A) is a trap. It’s just the number in the problem added together (2 + 9 + 7 + 15).

(B) CORRECT.

(C) is a trap for those who stop at 630 and don’t remember that it’s just a step on the way to finding the individual number of packets each person has.

(D) is the sum of the packets if starting with 1890 instead of 630 as the common multiple. 2k = 1890 (k = 945). 9m = 1890 (m = 210). 7s = 1890 (s = 270). 15b = 1890 (b = 126). Sum: 945 + 210 + 270 + 126 = 1551.

(E) is another possible multiple of 2, 9, 7 and 15. (2)(9)(7)(15) = 1890. But it’s not the least common multiple. It’s the trap for those who don’t omit a redundant 3, and who also forget to finish the solving. (The value of 2k = 9m = 7s = 15b is not the answer, k + m + s + b is the answer.)

The correct answer is B.

Woo-hoo! But I actually think my own method, with the compound ratio, is quicker.


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this week's MGRE Math Beast Challenge

From here:

Abe, Beata, Cruz, and Dion each collect a different type of condiment packet, and each person only collects one type. Twice the number of ketchup packets in Abe’s collection is 9 times the number of mustard packets Beata has, 7 times the number of soy sauce packets Cruz has, and 15 times the number of barbecue sauce packets possessed by Dion. If each collector owns at least one packet and only whole packets, what is the fewest possible number of packets owned by all four people?

(select only one)

A. 33
B. 517
C. 630
D. 1551
E. 1890

Go to it! My own answer will appear in the comments. Hint: the easiest way to solve this problem is probably via compound ratios. Ever worked with those? I started using them only a few months ago myself! Compound ratios basically look like multi-tiered fractions, because that's exactly what they are. Here's an example of such ratios in action:

In Arkansas, for every 2 snakes there are 7 gerbils; for every gerbil there are 5 mice. On Farmer Brown's many-acred property, there are 6125 mice. How many snakes are on Farmer Brown's property?

Start building a compound ratio by first assembling the data you have:

Let snakes = s; let gerbils = g; let mice = m.

s : g : m

2 : 7 : x

y : 1 : 5

Solve for x by making a proportion: 7/x = 1/5. X therefore equals 35.

s : g : m

2 : 7 : 35

No need to solve for y! We now have our compound ratio, and we can derive other ratios from the above information. To wit:

s/g = 2/7 (given)
g/m = 7/35 = 1/5 (given)
s/m = 2/35 (we'll need this info)

Now that we know the basic ratio of snakes to gerbils to mice, we can apply the compound ratio to the problem.

s : g : 6125

2 : 7 : 35

The snakes-to-mice ratio is 2 : 35. Set up a proportion:

2/35 = s/6125

Solve for s:

35s = 6125•2

s = (6125•2)/35 = 350

Farmer Brown's looking at 350 snakes on his property.

And that's how compound ratios work!



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rock and roll! got last week's MGRE problem correct!

I admit I felt shaky about my solution to last week's MGRE Math Beast Challenge problem, but I turned out to be correct. Here's MGRE's explanation:

Let’s put the words into equation form:

(Total Income – Exclusion)(Tax Rate) = Income Tax.

The question asks about total income, so we’ll solve the equation for Total Income:

Total Income – Exclusion = [Income Tax / Tax Rate]
Total Income = [Income Tax / Tax Rate] + Exclusion

To maximize total income, we’ll minimize Tax Rate (smaller denominator→larger value) and maximize the Exclusion:

Maximum Total Income = [$8700/0.15] + $9800 = $58,000 + $9,800 = $67,800.

To minimize total income, we’ll maximize Tax Rate (larger denominator→smaller value) and minimize the Exclusion:

Minimum Total Income = [$8700/0.35] + $5200 = $24,857.14 + $5,200 = $30,057.14.

The correct answers are C, D, E, and F.

The letters C, D, E, and F correspond to the values I had selected. MGRE arrived at the exact same range that I had arrived at, too: roughly $30,057 at the bottom end, and $67,800 at the top end.

Triomphe!


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this week's MGRE Math Beast Challenge

From here:

For Jack, income tax is between 15 and 35 percent of total income after an “exclusion” amount has been subtracted (that is, Jack does not have to pay any income tax on the exclusion amount, only on the remainder of his total income). If the exclusion amount is between $5200 and $9800, and Jack’s income tax was $8700, which of the following could have been Jack’s total income?

(Choose all that are appropriate)

$13,000

$23,100

$33,200

$43,300

$53,400

$63,500

$73,600

Go to it! I'll leave my response in the comments.


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solution to last week's MGRE Math Beast Challenge

From MGRE, this explanation for why the answer to last week's Math Beast Challenge is indeed (C), 36:

In situations in which two people, animals, cars, etc. are traveling on a straight line:

ADD the rates when moving in opposite directions.
SUBTRACT the rates when moving in the same direction.

In this case, if the prey is running at 40 kph and the predator is chasing at 48 kph, then the predator is catching up at a rate of 8 kph.

Since the distance between the two is 80 meters, we can simply use the formula Rate × Time = Distance.

However, note that the distance is in METERS and the rate is in KILOMETERS per hour. We will have to convert. Let’s use 1 kilometer = 1,000 meters to make a proportion:

80 = 1,000x
0.08 = x

Thus, the distance = 0.08 kilometers.

From Rate × Time = Distance,
8t = 0.08
t = 0.01

Thus, the time is 0.01 or 1/100 HOURS, but we need our answer in SECONDS. 1/100 of an hour is 60/100 of a minute, which is 3/5 of a minute, which is 36 seconds.

Or, make a proportion:

x = (0.01)(3600)
x = 36

The correct answer is C.

This was exactly the solution as my friend Charles described it. Bravo!


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yes! I never tire of being right!

I always get this nerdy feeling of accomplishment when I manage to get a Manhattan GRE Math Beast Challenge problem correct. My answer to last week's challenge was 17.473 (see the comments), and that's the correct answer. MGRE's explanation:

A ratio of one value to another value should not have units (i.e. we should not need to specify a currency unit, such as dollars), as both values can be expressed in terms of dollars and the units should cancel. However, both face value and intrinsic value must be expressed in terms of the same currency; Australian dollar unit doesn’t cancel the U.S. dollar unit. We’ll use USD to signify U.S. dollars and AUD to signify Australian dollars.

Putting both values in terms of U.S. dollars:
Intrinsic value = (1 troy ounce of gold)(1775.30 USD/troy ounce of gold) = 1775.30 USD
Face Value = (100 AUD)(1.016 USD/1 AUD) = 101.60 USD
Ratio = Intrinsic/Face = 1775.30/101.60 = 17.4734252 {Use the calculator!}

Rounded to three decimal places, the correct answer is 17.473.

On any problem that requires converting from one unit of measure to another, first think about what we want to cancel. When converting from AUD to USD, we want to cancel AUD units out. Therefore, multiply by a term that has AUD in the denominator. But we must always multiply by 1, or else we’ll change the value, so the top and bottom of any fraction multipliers must be equal. We were told that 1 AUD = 1.016 USD, so (1.016 USD/1 AUD) equals 1 and also serves to cancel the AUD units in the face value calculation above.

Not to be currency biased, we could have solved in terms of AUD:
Intrinsic value = (1 troy ounce)(1775.30 USD/troy ounce)(1 AUD/1.016 USD) = 1747.34252 AUD {Use the calculator!}
Face Value = 100 AUD
Ratio = Intrinsic/Face = 1747.34252/100 = 17.4734252

Rounded to three decimal places, the correct answer is 17.473.

Woo-hoo!

You know... if you've been ignoring these math problems when I put them up every Tuesday, I really encourage you to try them. Are you afraid to be wrong? What silliness! We don't learn by avoiding risks! And hell, I'm taking as much of a risk as any of you, since I don't know the official answer to any given problem until the following week.

So what's holding you back? Give these problems a try!


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