Rational Numbers and Proportional Reasoning: Test Review

Everything for the Unit 2 test on one page: the key facts from all 5 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Integers: Location, Direction and Change

Integers: $\dots, - 2, - 1, 0, 1, 2, \dots$ ; zero is neither positive nor negative.
On a number line the number further right is greater, so $- 2 > - 7$ .
Change $=$ final $-$ initial; a positive change is an increase, a negative change is a decrease.
Subtracting a negative is adding its opposite: $a - (- b) = a + b$ .
One negative sign makes a fraction negative: $\frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b}$ ; two negatives make it positive: $\frac{- a}{- b} = \frac{a}{b}$ (for $b \neq = 0$ ).
Rate of change $= \frac{change}{time taken}$ ; its sign tells you the direction.

Fractions, Unit Fractions and Measuring

A unit fraction is $\frac{1}{n}$ ; the fraction $\frac{a}{n}$ is $a$ copies of $\frac{1}{n}$ .
For positive unit fractions, a larger denominator means a smaller piece: $\frac{1}{16} < \frac{1}{8}$ .
Equivalent fractions: $\frac{a}{b} = \frac{a \times k}{b \times k}$ for any $k \neq = 0$ .
A typical imperial ruler has 16 gaps per inch; read a length by counting sixteenths past the last inch mark, then simplify.
1 in $= 2.54$ cm exactly; 1 cup is usually taken as 250 mL in Canadian recipes.

Operations with Positive and Negative Fractions

Add or subtract: $\frac{a}{d} \pm \frac{b}{d} = \frac{a \pm b}{d}$ , after rewriting with a common denominator $d$ .
Multiply: $\frac{a}{b} \times \frac{c}{d} = \frac{a c}{b d}$ (cancel common factors first).
Divide: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ , for $c \neq = 0$ .
Mixed number to improper fraction: $p \frac{q}{r} = \frac{p r + q}{r}$ ; and $- p \frac{q}{r} = - \frac{p r + q}{r}$ .
Sign rules for multiplying and dividing: same signs give a positive result, different signs give a negative result.

Ratios, Rates and Proportions

Equivalent ratios: $a : b = (ka) : (kb)$ for any $k \neq = 0$ .
Unit rate $= \frac{amount}{number of units}$ ; the lowest unit price is the best buy.
A proportion states two ratios are equal: $\frac{a}{b} = \frac{c}{d}$ .
Cross-multiplying: if $\frac{a}{b} = \frac{c}{d}$ (with $b, d \neq = 0$ ), then $a d = b c$ .
Map scale $1 : n$ : real distance $=$ map distance $\times n$ (then convert units).

Percentages: Increase, Decrease, Taxes and Discounts

$r %$ of an amount $= \frac{r}{100} \times$ amount.
Percent change $= \frac{new - old}{old} \times 100%$ (old value not zero; positive is an increase, negative a decrease).
Increase by $r %$ : multiply by $1 + \frac{r}{100}$ . Decrease by $r %$ : multiply by $1 - \frac{r}{100}$ .
Ontario HST: total $=$ price $\times 1.13$ ; pre-tax price $=$ total $\div 1.13$ .
Successive percent changes combine by multiplying the multipliers, not by adding the percents.

Where marks get lost

Saying $- 7 > - 2$ because 7 is bigger than 2. On the number line $- 2$ is further right, so $- 2 > - 7$ . (Integers: Location, Direction and Change)
Subtracting in the wrong order. A temperature going from $3^{\circ} C$ to $- 5^{\circ} C$ has change $- 5 - 3 = - 8$ , not $3 - (- 5) = + 8$ ; the second gives the right size but says it got warmer. (Integers: Location, Direction and Change)
Treating $5 - (- 14)$ as $5 - 14 = - 9$ . Subtracting a negative means adding: $5 + 14 = 19$ . (Integers: Location, Direction and Change)
Thinking $\frac{- 6}{- 8}$ is negative because it "has negatives in it". Two negative signs cancel, so it equals $\frac{3}{4}$ . (Integers: Location, Direction and Change)
Giving a rate without its sign. A level that drops 6 cm in 4 days changes at $- 1.5$ cm per day; writing $1.5$ loses the fact that it is falling. (Integers: Location, Direction and Change)
Thinking $\frac{1}{16}$ is bigger than $\frac{1}{8}$ because 16 is bigger than 8. More pieces means smaller pieces. (Fractions, Unit Fractions and Measuring)
Counting tick marks instead of gaps. From the 1-inch mark to the third tick after it is 3 gaps, i.e. $\frac{3}{16}$ in; counting the starting tick as well gives 4. (Fractions, Unit Fractions and Measuring)
Reading every tick as an eighth. Many rulers stop at sixteenths, so the short tick right after an inch mark is $\frac{1}{16}$ , not $\frac{1}{8}$ . (Fractions, Unit Fractions and Measuring)
Changing only the numerator or only the denominator: $\frac{3}{4}$ is not equal to $\frac{6}{4}$ . Multiply both by the same number. (Fractions, Unit Fractions and Measuring)
Converting $5 \frac{3}{8}$ in as $5.38 \times 2.54$ or, worse, $5.3 \times 2.54$ . Write $\frac{3}{8} = 0.375$ exactly, and round only at the end. (Fractions, Unit Fractions and Measuring)
Adding the denominators: $\frac{1}{3} + \frac{1}{4} = \frac{2}{7}$ is wrong. Rewrite with denominator 12 first: $\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$ . (Operations with Positive and Negative Fractions)
Reading $- 2 \frac{1}{3}$ as $- 2 + \frac{1}{3}$ . The sign covers the whole number: $- 2 \frac{1}{3} = - \frac{7}{3}$ . (Operations with Positive and Negative Fractions)
Flipping the wrong fraction when dividing. In $\frac{2}{3} \div \frac{4}{5}$ only the second fraction flips: $\frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$ . (Operations with Positive and Negative Fractions)
Multiplying mixed numbers part by part: $2 \frac{1}{2} \times 3 \frac{1}{3}$ is not $6 \frac{1}{6}$ . Convert first: $\frac{5}{2} \times \frac{10}{3} = \frac{50}{6} = 8 \frac{1}{3}$ . (Operations with Positive and Negative Fractions)
Losing a sign: $(- \frac{3}{4}) (- \frac{8}{9})$ is positive, $\frac{2}{3}$ , because two negative factors give a positive product. (Operations with Positive and Negative Fractions)
Reversing the order of a ratio. If the ratio of concentrate to water is $1 : 3$ , then water to concentrate is $3 : 1$ , not $1 : 3$ . (Ratios, Rates and Proportions)
Setting up a proportion with mismatched order, such as $\frac{cups}{cookies} = \frac{cookies}{cups}$ . Keep the same quantity on top on both sides. (Ratios, Rates and Proportions)
Assuming the biggest package is always the best buy. Compare unit prices: in the cereal example the middle box was cheapest per 100 g. (Ratios, Rates and Proportions)
Comparing unit prices in different units, such as dollars per 100 g against dollars per kg. Convert to one unit first. (Ratios, Rates and Proportions)
Forgetting to convert map units: 3.2 cm on a $1 : 50000$ map is $160000$ cm, which is 1.6 km, not 160 000 km. (Ratios, Rates and Proportions)
Adding percents for successive changes: 25% off then 13% tax is not 12% off. Multiply the multipliers: 0.75 × 1.13 = 0.8475. (Percentages: Increase, Decrease, Taxes and Discounts)
Thinking +10% then −10% returns to the start. The second percent is taken of a different (larger) amount, so the result is 1% lower. (Percentages: Increase, Decrease, Taxes and Discounts)
Dividing by the new value in a percent change. Always divide by the original (old) value. (Percentages: Increase, Decrease, Taxes and Discounts)
Taking 13% off a tax-included total to find the pre-tax price. Divide the total by 1.13 instead. (Percentages: Increase, Decrease, Taxes and Discounts)
Using 0.13 as the multiplier for adding tax. 0.13 gives the tax alone; 1.13 gives price plus tax. (Percentages: Increase, Decrease, Taxes and Discounts)

Practice test

25 questions from across the unit. Each one shows the lesson it comes from.

Integers: Location, Direction and Change
1
Which of these numbers is the greatest?
Integers: Location, Direction and Change
2
A hiker starts at the shore of a lake 40 m below sea level and climbs to a lookout 25 m above sea level. What is her change in elevation?
m
Integers: Location, Direction and Change
3
Put these numbers in order from least to greatest.
1. 12
2. 20
3. 3−1/2
4. 4−8
5. 5−3.5
Integers: Location, Direction and Change
4
Write $\frac{- 18}{- 24}$ as a decimal.
Integers: Location, Direction and Change
5
What is $∣ - 23∣$ , the absolute value of $- 23$ ?
Fractions, Unit Fractions and Measuring
6
Which of these is a unit fraction?
Fractions, Unit Fractions and Measuring
7
The end of a nail is 6 sixteenth-gaps past the 0 mark on an imperial ruler. What is its length in simplest form?
Fractions, Unit Fractions and Measuring
8
Match each fraction of an inch with the same length in sixteenths.
1/2
1/4
3/8
1/8
Fractions, Unit Fractions and Measuring
9
Write 10/16 in simplest form (type it like 3/4).
Fractions, Unit Fractions and Measuring
10
A 9/16 inch wrench is how many inches wider than a 1/2 inch wrench? Give the answer as a fraction or a decimal.
in
Operations with Positive and Negative Fractions
11
Calculate $\frac{1}{3} + \frac{1}{4}$ . Enter a fraction such as 5/6.
Operations with Positive and Negative Fractions
12
Calculate $- \frac{5}{6} \div \frac{5}{12}$ .
Operations with Positive and Negative Fractions
13
How many pieces $1 \frac{1}{3}$ ft long can be cut from an 8 ft board? (Ignore the width of the saw cut.)
pieces
Operations with Positive and Negative Fractions
14
Match each calculation with the best first step.
1/3 + 1/5
2/3 × 9/10
2/3 ÷ 4/5
2 1/3 − 1/2
Operations with Positive and Negative Fractions
15
What is the reciprocal of $- \frac{2}{5}$ ?
Ratios, Rates and Proportions
16
Solve the proportion $\frac{x}{15} = \frac{4}{6}$ .
Ratios, Rates and Proportions
17
On a map with scale $1 : 50000$ , a trail measures 3.2 cm. How long is the real trail, in kilometres?
km
Ratios, Rates and Proportions
18
Write the ratio 18 : 24 in simplest form (type it like 2:5).
Ratios, Rates and Proportions
19
Match each example with what it is.
3 cups of flour to 2 cups of sugar
2.50 dollars per kilogram
1 cm on the map is 500 m on the ground
x/12 = 5/8
Ratios, Rates and Proportions
20
A car uses 7.5 L of fuel per 100 km. How many litres does it use on a 340 km trip?
L
Percentages: Increase, Decrease, Taxes and Discounts
21
What is 15% of 60?
Percentages: Increase, Decrease, Taxes and Discounts
22
A price is increased by 20%, then the new price is decreased by 20%. What is the overall change?
Percentages: Increase, Decrease, Taxes and Discounts
23
Which multiplier gives the sale price after a 35% discount?
Percentages: Increase, Decrease, Taxes and Discounts
24
Match each change with its multiplier.
13% increase
13% decrease
30% decrease
100% increase
Percentages: Increase, Decrease, Taxes and Discounts
25
After a 20% discount, a pair of boots costs $64. What was the original price?

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…