This is a chem practice question from AAMC.
I won't lie. I'm horrible at math. I'm so confused how to do this math without a calculator. I knew what I needed to do, but just don't know how without a calculator.
This was the answer to the question:
"The solution contains 7.15 g Na2CO3×10H2O. Dividing by the molar mass will give the number of moles: (7.15 g)/(286.14 g/mol) = 0.0250 mol. Because each mole of Na2CO3×10H2O contains 2 mol Na+, there is 0.0500 mol Na+. Using Avogadro’s number, the number of sodium ions is (0.0500 mol)(6.02 × 1023 ions/mol) = 3.01 × 1022 ions. Thus, B is the best answer."
MCAT Math

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Re: MCAT Math
Hey there!
A good first step would probably be to forget everything you've learned about arithmetic. Unlike middle school, you will need to show no work, and not go about anything in a particular prescribed way. As long as you can describe the system and relations to yourself, and do a few very rudimentary operations, you will be perfectly set for all MCAT math. And usually you will want to arrive at an answer within the correct order of magnitude and units.
There are 3 arithmetic skills you need for MCAT quantitative problems:
1. Multiplying exponents and adjusting exponents in SI notation for easier manipulation
2. Rounding coefficients to make them more 'tractable' (read: be lazy and make the problem easier)
3. Dimensional analysis, just in case. This is where you forget about numbers entirely and only check what units you are producing.
(Note that logarithms and their use are also tested on the MCAT, but usually far less frequently and intensively  after you are fluent with exponents and SI notation, definitely take a look at the rules for logarithms)
#0 Exponents and their notation:
Not so much a skill but some necessary knowledge. The exponent just means "multiply this by itself that many times". I assume you are familiar with that. Negative exponents just mean 1/[whatever we wrote]. So 7^2 is just 1/7^2 = 1/49 (or to use our rounding skillset: ~1/50 = 0.02 = 2 ^ 10^2)
#1 Multiplying exponents:
Multiplying exponents adds them up (2^2 * 2^3 = 2^5). Exponentiating exponents multiplies them (2^2^3 = 2^6 = 64). For division, just flip the whole thing and make it multiplication. Together with your rules for exponent notation, an example like "divide by 10^12" becomes "multiply by 10^12").
#2 Rounding your coefficients:
Move things around so they become easy multiples of each other. If presented with a problem that requires 14/3, just turn it into 15/3 and be done with it. Ideally keep track of whether your answer over or underestimates. So in this case we could have our answer be "3 or a little less than that". This also works for otherwise unsightly fractions: Rather than computing 2.9/15, round to 3/15 and then just compute 30/15 and divide your answer by 10 (so 0.2).
#3 Dimensional analysis as a sanity check:
This is a really fancy word for a simple process. You make a chain of multiplication of all your base units and simplify maximally and see what units pop out at the end. If you are determining an energy but units of power pop out, you have made a mistake somewhere. I won't be going into this much more than that right here, because it is hardly helpful for this problem.
So for this problem:
I understand that this is a lot of words, but the key point is that you just want to be extremely familiar with a few rules of arithmetic for the MCAT. You aren't expected to be some inhuman mental computer. Let me know if this helps  and I'm happy to write up the rules of logarithms too, although frankly most of them are hardly tested on the MCAT (most emphasis is usually placed on just understanding the logarithmic scale).
A good first step would probably be to forget everything you've learned about arithmetic. Unlike middle school, you will need to show no work, and not go about anything in a particular prescribed way. As long as you can describe the system and relations to yourself, and do a few very rudimentary operations, you will be perfectly set for all MCAT math. And usually you will want to arrive at an answer within the correct order of magnitude and units.
There are 3 arithmetic skills you need for MCAT quantitative problems:
1. Multiplying exponents and adjusting exponents in SI notation for easier manipulation
2. Rounding coefficients to make them more 'tractable' (read: be lazy and make the problem easier)
3. Dimensional analysis, just in case. This is where you forget about numbers entirely and only check what units you are producing.
(Note that logarithms and their use are also tested on the MCAT, but usually far less frequently and intensively  after you are fluent with exponents and SI notation, definitely take a look at the rules for logarithms)
#0 Exponents and their notation:
Not so much a skill but some necessary knowledge. The exponent just means "multiply this by itself that many times". I assume you are familiar with that. Negative exponents just mean 1/[whatever we wrote]. So 7^2 is just 1/7^2 = 1/49 (or to use our rounding skillset: ~1/50 = 0.02 = 2 ^ 10^2)
#1 Multiplying exponents:
Multiplying exponents adds them up (2^2 * 2^3 = 2^5). Exponentiating exponents multiplies them (2^2^3 = 2^6 = 64). For division, just flip the whole thing and make it multiplication. Together with your rules for exponent notation, an example like "divide by 10^12" becomes "multiply by 10^12").
#2 Rounding your coefficients:
Move things around so they become easy multiples of each other. If presented with a problem that requires 14/3, just turn it into 15/3 and be done with it. Ideally keep track of whether your answer over or underestimates. So in this case we could have our answer be "3 or a little less than that". This also works for otherwise unsightly fractions: Rather than computing 2.9/15, round to 3/15 and then just compute 30/15 and divide your answer by 10 (so 0.2).
#3 Dimensional analysis as a sanity check:
This is a really fancy word for a simple process. You make a chain of multiplication of all your base units and simplify maximally and see what units pop out at the end. If you are determining an energy but units of power pop out, you have made a mistake somewhere. I won't be going into this much more than that right here, because it is hardly helpful for this problem.
So for this problem:
Code: Select all
Dimensional: g / (g/mol) = g * mol/g = mol > We are good to go, this checks out
7.15 / 286.14 > (715 / 286) * 10^2 > Parenthesis are only here for clarity
700 / 280 = 2.5 > This should be pretty accurate, a slight underestimation. Doesn't matter if you use a rougher approach and get 2.42.6.
Intermediate answer: 2.5 * 10^2 moles
Adjust for 2 moles of sodium per molecule: 5.0 * 10^2 moles
Now multiply this all by 6*10^23 to get the number of atoms.
6 * 5 * 10^2 * 10^23 = 30 * 10^21 sodium ions > Remember, we can just add exponents and multiply coefficients. Barely an inconvenience!
Final answer: 3 * 10^22 sodium ions > Adjusted into standard SI (coefficient 10 times smaller, so the power of 10 becomes 1 larger)
I understand that this is a lot of words, but the key point is that you just want to be extremely familiar with a few rules of arithmetic for the MCAT. You aren't expected to be some inhuman mental computer. Let me know if this helps  and I'm happy to write up the rules of logarithms too, although frankly most of them are hardly tested on the MCAT (most emphasis is usually placed on just understanding the logarithmic scale).