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# High School: Number and Quantity

### Quantities HSN-Q.A.3

3. Choose a level of accuracy appropriate to the limitations on measurement when reporting quantities.

Timmy and Teddy are in kindergarten. They're best friends, which means, of course, that they sometimes argue like worst enemies. At the moment, they're arguing over who is the number one fan of the Chicago Bulls. Their conversation goes something like this:

Timmy: I'm the best fan. I'm a hundred times better than you.

Teddy: No, I'm the best fan. I'm a hundred million times better than you.

Timmy: Oh yeah? I'm a bazillion times better than you.

Teddy: Oh yeah, I'm infinity times better than you.

Let's not even address the fact that infinity, since it isn't a number, can't multiply anything. But let's talk about how either boy could prove his point. They're five. Neither one has season tickets or has even actually been to a game. Neither one is part of the Chicago Bulls' official fan site. Neither one could tell you the difference between a lay-up and a foul shot. Neither one could tell you any significant statistics. Neither one knows Dennis Rodman from Dennis the Menace.

In short, the odds are good that neither one has a very accurate appraisal of what it means to be "the number one fan of the Chicago Bulls," because neither has an accurate way to measure it.

It's the same in the world of real measurement. A measurement is only as accurate as its weakest partâ€”its "weakest link" in a manner of speaking. The level of accuracy in any measurement depends on both the methods used to obtain that measurement and the units being provided.

#### Drills

1. Kenny has a jar full of coins he's been saving, just like he's been doing every year. Which would provide Kenny with the most accurate idea of the amount of money he has in the jar?

Correct Answer:

Setting up a proportion between the height of the jar and the amount that last year's height yielded

Answer Explanation:

Of course, the most accurate way to determine the amount of money is to either count it or to let a machine at the bank count it. But if Kenny wants an approximation, choice (D) will give him the most accurate picture. All the other methods fail to differentiate between a jar that's mostly pennies and one that's mostly quarters. If Kenny's savings habits haven't changed, he can assume that the same amount of space in the jar yields about the same amount of money from year to year.

2. Charlie Brown has decided to run for Student Body President. Before he invests a lot of money in a campaign, he wants to determine what percent of the student body would vote for him over his rival, Peppermint Patty. Which would give him the most accurate idea?

Correct Answer:

Asking every third kid who enters the building one morning

Answer Explanation:

Both choices (A) and (B) represent a sample which might logically be skewed in favor of one or the other candidates. (Though, knowing Charlie Brown, we're guessing we know which candidate.) As far as choice (C) goes, Charlie Brown has 6 or 7 teachers. We aren't convinced that 6 or 7 teachers can accurately speak on behalf of the hundreds of kids involved in this election. But if Charlie Brown were to randomly survey a third of the student body, it should give him a pretty accurate idea of his chances.

3. In December, Phineas is trying to predict the cost of gasoline in his hometown for the summer. Which will give him the most accurate prediction?

Correct Answer:

Finding the cost on every Memorial Day in the past fifteen years

Answer Explanation:

To be honest, he should probably just guess "high" and then double it. But if he wants to predict the future, his best bet is to look at the past 15 years. Choices (A) and (C) are too limited, and choice (D) presumes that the attendant has any control or knowledge over the price.

4. Ferb is using a standard ruler to measure his height, to see if he's tall enough for the Sickenator Roller Coaster at his local amusement park. Which is the unit of measurement to which he can expect to be accurate?

Correct Answer:

Centimeter

Answer Explanation:

Assuming he doesn't count the height added on by his hair, most rulers are accurate to the nearest centimeter. A foot doesn't tell him much, and a yard will tell him even less. A normal ruler probably wouldn't be accurate enough to the nearest millimeter, but centimeters are a good bet.

5. Candace is on a diet, trying to impress her boyfriend with her healthy eating habits. She has a list of the number of Calories in each component of her meal. Which will probably be about the same as one of Candace's meals, Calorie-wise?

Correct Answer:

A quarter-pound burger without a bun

Answer Explanation:

An apple is hardly enough for a meal, and a diet soda should have zero calories. (If it doesn't, a serious complaint should be filed with the soda companies.) An order of fries could contain anywhere from five to five hundred fries. It's not specific enough to get a good estimate. A quarter-pounder without a bun gives a more accurate guess as to the Calorie-content of an actual meal.

6. Victor claims to have measured the thickness of a penny with a ruler. Which of the following is most likely the value Victor measured?

Correct Answer:

1.5 mm

Answer Explanation:

While (A) and (B) are both the correct measurements for the thickness of a penny, a simple ruler wouldn't be able to deliver that kind of precision. Typically, rulers have markings for each millimeter and each eighth or sixteenth of an inch. That means the most precise Victor can get is (C) or (D). Since (C) is clearly not the correct measurement, our answer (and Victor's answer, too) is (D).

7. Sandra has been practicing piano for years, and she's finally decided to buy one. Since it's such a big investment, she wants to find the average price of a piano to make sure she's not being overcharged. Which of the following would give Sandra the most accurate average?

Correct Answer:

Averaging the prices on Craigslist and the piano store down the street

Answer Explanation:

The first prices Sandra sees on Craigslist might not be an accurate representation of the many different kinds and prices of pianos available. Just the same, the piano store and manufacturing companies might be more expensive than some of the used pianos on Craigslist. The best solution is to average the used pianos on Craigslist with the newer ones in the store, or (B).

8. Olivia is training to run a marathon. She's pretty intense. If a marathon is about 26.2 miles and there are 63,360 inches in a mile, how many inches will Olivia be running?

Correct Answer:

1,660,000 in

Answer Explanation:

The key in this problem is the little word "about" before 26.2 miles. We can calculate the total number of inches and we'll end up with (A), so that's the answer, right? Nope. We only know the precision up to 0.1 or 0.2 miles. That's about 10,000 inches. That means that we can only be precise up to the 10,000-inch point. Answer (A) is too specific, and (B) and (D) are too general. Answer (C) is just right.

9. Alex weighs 168 pounds exactly. (Sure, it fluctuates from time to time, but as of this very moment, that's how much he weighs.) If 1 pound equals 2.2 kilograms, how many kilograms does Alex weigh right now?

Correct Answer:

369.6 kg

Answer Explanation:

When we multiply numbers, we have to consider how accurate we are. Since the problem tells us that Alex weighs exactly 168 pounds, we can assume it means 168.0 or 168.00 or 168.0000000 pounds. You can add as many zeros as you want after the decimal point for Alex's weight. It won't change it. When we multiply 168 by 2.2, we can take the answer to one number past the decimal place. (We're still limited by 2.2 kg per pound.) That means (A) is our most accurate answer.

10. Zoe is about 5 feet and 6 inches tall. There are 12 inches in a foot and 2.54 centimeters in an inch. How tall is Zoe in centimeters?

Correct Answer:

168 cm

Answer Explanation:

Since we know that Zoe is about 66 inches tall, the most precise we can be is up to the nearest inch. In other words, that means Zoe is about 167.64 cm, but we can't be that certain in our answer. The most certain we can be is within a range of about 5 centimeters. Answer (A) is a bit too general, and even (C) is too specific, so our best bet is (B).