Numerical Problem (Consumer Price Index) by how much has it risen between start of

[p1nkm4n]

Hi everyone! Been preparing for my GRE for a while and stuck with a few problems as I've been out of touch from Math since Year 2 of my degree, which was in 2013. I'll be posting each in separate threads. The problem is quoted below:
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Price of Soya Beans​

Year	Price in $/tonne (Dec 31st)	Consumer Price Index (Jan 1st - Dec 31st)
1	$340	+3.9%
2	$358	+4.2%
3	$348	+3.1%
4	$346	+2.5%
5	$359	+1.7%
6	$375	+2.1%

By how much has the Consumer Price Index risen between the start of Year 1 and the end of Year 6?

A. 11.90% B. 13.60% C. 15.40% D. 17.50% E. 18.80%

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While I could make some attempts on the other questions, however I am completely unaware of such problems related to consumer price index, as I've never done them before. Probably missing out a formula or concept here. Any help is highly appreciated. Thank you for reading!

 

[Dr.Peterson]

Hi everyone! Been preparing for my GRE for a while and stuck with a few problems as I've been out of touch from Math since Year 2 of my degree, which was in 2013. I'll be posting each in separate threads. The problem is quoted below:
​

Price of Soya Beans​

Year	Price in $/tonne (Dec 31st)	Consumer Price Index (Jan 1st - Dec 31st)
1	$340	+3.9%
2	$358	+4.2%
3	$348	+3.1%
4	$346	+2.5%
5	$359	+1.7%
6	$375	+2.1%

By how much has the Consumer Price Index risen between the start of Year 1 and the end of Year 6?

A. 11.90% B. 13.60% C. 15.40% D. 17.50% E. 18.80%

---

While I could make some attempts on the other questions, however I am completely unaware of such problems related to consumer price index, as I've never done them before. Probably missing out a formula or concept here. Any help is highly appreciated. Thank you for reading!

You don't really need to know anything about the consumer price index. What you do need to know is how percent increases combine.

One of the errors they expect you to make is just to add the percentage increases together, and give that as the answer. But each percent increase is relative to a different base, so you can't do that. What you can do is to find the cumulative increase by seeing a 3.9% increase, say, as a multiplication by 103.9% (1.039). Are you familiar with that idea?

 

[p1nkm4n]

You don't really need to know anything about the consumer price index. What you do need to know is how percent increases combine.

One of the errors they expect you to make is just to add the percentage increases together, and give that as the answer. But each percent increase is relative to a different base, so you can't do that. What you can do is to find the cumulative increase by seeing a 3.9% increase, say, as a multiplication by 103.9% (1.039). Are you familiar with that idea?

Honestly no, could you please elaborate more?

 

[Dr.Peterson]

Honestly no, could you please elaborate more?

Suppose that I raise the price on an item by 10%, and then raise it by another 25% after that. The total increase is not 10+25=35%.

Rather, a 10% increase makes the new price 100% + 10% = 110% of the original, and likewise a 25% increase makes it 125%. So the new price is 1.10 * 1.25 = 1.375 times the original, which is a 37.5% increase. This is more than 35% because the second increase added 25% of the already increased amount, not of the original.

 

[JeffM]

This is substantively the same answer as given by Dr. Peterson.

Let's say a store charges 20 dollars for an item The item is very popular so the store owner decides to raise the price by 10%.

What is the second price? Obviously it is 22 dollars.

\(\displaystyle 20.00 * (1 + 10\%) = 20.00 * (1 + 0.1) = 20.00 * 1.1 = 22.\)

The store owner finds that the second price did not decrease sales so she again decides to raise the price, but this time by 5%.

What is the third price?

\(\displaystyle 22.00 * (1 + 5\%) = 22.00 * (1 + 0.05) = 22.00 * 1.05 = 23.10.\)

You will get the wrong answer if you add the relative changes and multiply by the first price.

\(\displaystyle 20.00 * (1 + 10\% + 5\%) = 20.00 * (1 + 0.1 + 0.05) = 20.00 * 1.15 = 23.00 \ne 23.10.\)

Now, the error is small in this case, but the more relative changes are involved, the bigger the error becomes.

In technical language, the first and second relative price changes are relative to different bases. The 10% was relative to the price of 20. The 5% was relative to the price of 22. The prices are the bases for the relative measurement of the change in price. Each change was measured relative to its own base, and obviously 20 and 22 are different numbers.

This issue of relative change with respect to changing bases is what is behind all of the math involving compound interest and everything else (such as population growth) with a similar mathematical structure.