Rational numbers in real situations

When you reach for it

Many measurements are made from a reference point and can land on either side of it. A river gauge gives the water’s height above or below flood stage, the level at which the river starts to spill over its banks. A golf score is counted over or under par, the number of strokes the course is expected to take. A share price moves up or down from where it closed the day before. A runner’s time for each kilometer is over or under her target. Clocks around the world are set ahead of or behind one reference time. In each case a single signed number says two things at once: how far from the reference point, and on which side.

The arithmetic is already in hand: adding and subtracting and multiplying and dividing signed fractions and decimals. What this lesson adds is the move between the story and the numbers: deciding what zero stands for, turning words such as “below”, “under”, “fell” and “slow” into signs, choosing the operation the question asks for, and reading the sign of the answer back as words.

The procedure

The lesson is taught in two parts. The first sets up signed quantities and works with sums, differences and order. The second works with products, quotients and averages, and with problems of several steps.

Part 1: zero, direction and change

Choose a zero and a positive direction, and say what they are. A signed number is a position measured from a reference point, the way negative numbers mark positions on either side of 0 on a number line. So the first decision is what 0 stands for (flood stage, par, the target pace, last night’s closing price) and which side counts as positive. Usually the choice is natural: up, above, ahead and gain are positive; down, below, behind and loss are negative. A reading 0.6 meters below flood stage is then -0.6, and \displaystyle \frac{3}{4} of a meter above it is \displaystyle \frac{3}{4}. The choice is not forced. Counting downward as positive would flip every sign in the problem, every answer would flip with it, and the answers read back into words would be the same. What matters is to make the choice once and keep to it.

Order on the number line is order in the story. With up as positive, a greater number is a higher level: -0.6 > -1.25 says that water at -0.6 meters stands higher than water at -1.25 meters, even though 0.6 is less than 1.25. The absolute value answers a different question, how far from the reference point, on whichever side: |-1.25| = 1.25, so the reading -1.25 is further from flood stage than -0.6 is. “Which is higher?” is answered by the order of the numbers; “which is closer to flood stage?” is answered by their absolute values.

A change is added to the starting value. If a quantity starts at a and changes by c, it ends at a + c. On the number line a + c is the point |c| away from a, up (or to the right) when c is positive and down (or to the left) when c is negative. Several changes in a row add up to one net change. Since addition is commutative and associative, the changes may be added in any order and grouping, and putting the rises together and the falls together is often quickest. Two changes that are opposites add to 0, and a net change of 0 means the quantity ends where it began.

The change from one value to another is later minus earlier. Since \text{start} + \text{change} = \text{end}, the change is

A positive change is a rise and a negative change is a fall. Subtracting the other way round gives the opposite number, which describes the same trip in reverse. If the question asks only how far apart two values are, with no direction, the answer is the distance |a - b|, which is the same either way round.

Part 2: rates, averages and several steps

Equal changes repeated make a product. A change of c repeated n times is a change of n \cdot c. When something changes at a steady rate of r per hour (or per day, or per kilometer), the change over a time t is r \cdot t. The sign rules for multiplication say the right thing: a negative rate over a positive length of time gives a negative change, a fall. A level falling 0.2 meters an hour has rate -0.2, and over 3 hours it changes by (-0.2)(3) = -0.6 meters.

A rate is a quotient. The same relation run backwards gives r = \text{change} \div t, a rate found by dividing a change by the time it took, and t = \text{change} \div r, the time a given change needs.

An average of signed values is their sum divided by how many there are. The Mean of -3, 1 and -4 is

The mean of signed measurements is itself a signed measurement from the same zero: a mean of -2 strokes against par means 2 strokes under par on average. The mean of integers need not be an integer, which is one way fractions and decimals turn up even when every measurement is whole.

Several steps: one expression, the right form, an estimate. A problem with several steps is easier to follow, and to check, when it is written as one expression before anything is worked out. Decimals suit money and measurements already given as decimals; a fraction keeps a result exact when its decimal would not end, as the decimals for \displaystyle \frac{1}{3} and \displaystyle \frac{1}{6} do not. Then round the numbers and estimate the answer: an exact result far from the estimate means a step has gone wrong, such as a decimal point in the wrong place or a minus sign dropped. Read the answer back into words, sign included, and ask whether it fits the story.

Worked examples

Examples 1 to 3 belong to the first part. Examples 4 to 7 belong to the second.

Boundaries

A signed number needs a stated zero, unit and direction. “-0.6” alone says nothing about a river; “-0.6 meters against flood stage, up positive” does. Two signed numbers can be added, subtracted or compared only when they are measured from the same zero, in the same unit, with the same direction counted positive. A gauge reading against flood stage and a height against sea level cannot be subtracted until one is rewritten from the other’s zero.

A rate times a time assumes the rate stays steady. The watch in Example 4 and the river in Example 5 were given steady rates. A real river does not fall at one rate for ever, so a prediction made with one fixed rate is only an approximation, and it gets less reliable the further it reaches.

A time to reach a value exists only when the change needed and the rate have the same sign. If the signs differ, the quantity is moving away from the value, the quotient is negative, and with a steady rate it says how long ago the quantity was at that value. With a rate of 0 there is no quotient at all: a level that is not changing never reaches a different level.

Rounding comes last. Money is paid in whole cents, but rounding before the last step can move the result. Keep values such as \displaystyle -\frac{31}{60} exact until the end, and round once.

The answer must fit the story. A count of strokes, rounds or days, or a length of time from now, cannot be negative; a calculation that gives one has subtracted the wrong way round or used the wrong operation.

Where this goes next

Writing a story as one expression is the first step to writing it as an equation. The sixth golf round in Example 6 and the last kilometer in Example 3 were equations solved in the head, and harder ones are solved the same way on paper. A steady rate of change, like the falling river’s, is the Slope of a straight-line graph of level against time, and a falling level gives a negative slope. When the rate is not steady, the change over a stretch of time divided by its length is an average rate of change. A signed change measured as a share of the starting value is a percent change, which is how a change in a share price is often given. And once the mean of signed measurements is known, the next question is how spread out they are around it, which is the mean absolute deviation.

Exercises

Part 1: signed quantities and order

1. Choose a zero and a positive direction, and write each as a signed number. (a) A share price fell 0.65 dollars today. (b) A river is \displaystyle 2\frac{3}{4} meters above flood stage. (c) A runner’s kilometer was 1.8 seconds faster than her target, with time over the target counted positive. (d) A watch is 12.5 seconds fast.
2. A tide table gives the height of the sea above (positive) or below (negative) a fixed level called chart datum. The low tides on four days were -0.3, 0.15, -0.45 and -0.1 meters. (a) Order them from lowest to highest. (b) Which low tide was furthest from chart datum, and was it above or below?
3. A metal part should measure 25 millimeters across. Each part’s error is its measured width minus 25 millimeters, and five parts had errors of +0.03, -0.05, -0.02, +0.06 and -0.04 millimeters. A part is accepted if its error is no more than 0.04 millimeters from 0, on either side. (a) Which parts are accepted? (b) How wide was the narrowest part?

Part 1: change, distance and zero sums

4. A share’s price changed by -0.38, +1.15, -0.92 and +0.40 dollars on four days in a row. It started at 12.70 dollars. Find the net change and the final price.
5. One morning the tide went from a low of -0.35 meters against chart datum to a high of 2.4 meters. Find the change. It then fell to a low of 0.2 meters. Find that change too.
6. Mumbai is \displaystyle 5\frac{1}{2} hours ahead of UTC and Honolulu is 10 hours behind it. (a) How many hours ahead of Honolulu is Mumbai? (b) When it is 9{:}00 p.m. on Monday in Honolulu, what day and time is it in Mumbai?
7. A runner’s target is 5 minutes per kilometer for 4 kilometers. Her first three kilometers were +3.6, -1.4 and +0.9 seconds against the target. How long must her last kilometer take for her to finish in exactly 20 minutes?
8. At 7 a.m. a river gauge read -0.85 meters against flood stage. During the day the level changed by +1.2, then +0.65, then -0.3 meters. What was the final reading, and was the river above or below flood stage?
9. After two rounds of a golf tournament Femi is 5 under par and Gita is 2 over par, and in golf the lower score leads. Write both scores as signed numbers, say who leads, and find by how many strokes.
10. A carpenter fixes three shelf brackets against a level pencil line. Measured from the line, with up positive, they sit at \displaystyle \frac{3}{16}, \displaystyle -\frac{1}{8} and \displaystyle -\frac{5}{16} inches. How far is the highest bracket above the lowest?

Part 2: rates and averages

11. A clock gains 0.75 seconds a day. (a) How far out is it 28 days after being set right? (b) After how many days is it a full minute fast?
12. Over a week of 5 trading days, the daily changes in a share’s price had a mean of -0.34 dollars. (a) What was the net change in the price over the week? (b) By how much did the value of 60 of these shares change?
13. A river gauge reads 0.6 meters above flood stage and the level is falling steadily by 0.15 meters an hour. (a) In how many hours will it reach flood stage? (b) What will it read 10 hours from now, if the rate holds?
14. A tide rose from -0.4 meters against chart datum to 1.85 meters over \displaystyle 6\frac{1}{4} hours. Taking the rise as steady, find its rate in meters per hour.
15. On a course with par 71, a golfer’s scores against par in four rounds were -2, -5, +1 and -1. Find her mean score against par and her mean number of strokes per round.
16. A runner aims to run 10 kilometers at 4 minutes 30 seconds per kilometer. Her kilometer times have a mean of +0.85 seconds against that target. What is her total time?
17. Watch A loses 1.2 seconds a day and watch B gains 0.8 seconds a day. Both are set right at the same moment. Three weeks later, how fast or slow is each, and how far apart are they?

Part 2: several steps

18. Kwame buys 80 shares at 31.25 dollars each. Over three weeks the price changes by +1.40, -3.15 and -0.60 dollars. (a) What is the price now? (b) By how much has the value of his shares changed? (c) Check (b) with an estimate.
19. A drone takes off from a pad on top of a cliff and hovers 12.5 meters above the pad. It then descends at 1.5 meters per second for 11 seconds, passing the cliff edge and dropping down beside the cliff face. (a) Where is it now, measured from the pad? (b) Climbing at 0.8 meters per second, how long does it take to get back level with the pad?
20. A floor is checked against a target level at four points. Measured from the target, with up positive, the floor sits at \displaystyle -\frac{3}{8}, \displaystyle \frac{1}{4}, \displaystyle -\frac{1}{16} and \displaystyle \frac{5}{16} inches. Find the mean, and say whether the floor is on average above or below the target.

Reasoning

21. A river gauge reads 0.6 meters above flood stage and the level is rising steadily by 0.1 meters an hour. Work out the change needed to reach flood stage divided by the rate. Explain what the sign of the answer means, and why the water will not reach flood stage while it keeps rising.
22. A golfer plays n rounds on a course with par p, and her scores against par are d_1, d_2, \ldots, d_n. Explain why her mean number of strokes is p plus her mean score against par.
23. Suppose the runner in Example 3 had counted time under her target as positive instead. Say what each number in the example and its answers becomes, and explain why the answers read in words stay the same.
24. Without working it out exactly, decide whether 48 \cdot (-2.95) is nearer -100, -150 or -200, and explain how such an estimate shows that an answer of -14.16 cannot be right.