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Operations with Signed Rational Numbers: Core practice

10 practice problems for this lesson. Work on paper, use hints when you need them, and check the answer or the full solution when you are ready.

Difficulty: Core (core-course level)

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Problem 1 of 10

Problem 1 Three numbers, written with integers

Write each of these numbers as a fraction $\frac{a}{b}$ , where $a$ and $b$ are integers and $b$ is not zero: $-0.35$ , $-2\tfrac{3}{4}$ and $0$ . Give each fraction in lowest terms with a positive denominator.
1. Hint 1
  A decimal that ends counts tenths, hundredths or thousandths, and its number of decimal places tells you which.
2. Hint 2
  The minus sign in front of a mixed number belongs to the whole of it, the whole-number part and the fraction part together.
3. Hint 3
  Zero parts taken out of any nonzero number of parts is still zero, so think about which denominator is the smallest positive one.
Answer

$-0.35 = \frac{-7}{20}$ , $-2\tfrac{3}{4} = \frac{-11}{4}$ and $0 = \frac{0}{1}$ . Writing the minus sign in front instead, as $-\tfrac{7}{20}$ and $-\tfrac{11}{4}$ , names the same numbers.

Full solution

The decimal $0.35$ has two decimal places, so it counts hundredths:
$0.35 = \frac{35}{100}.$
The greatest common factor of $35$ and $100$ is $5$ , and dividing both by it gives $\frac{7}{20}$ .
$-0.35 = -\frac{35}{100} = \frac{-7}{20}$

The minus sign covers the whole mixed number, so $-2\tfrac{3}{4}$ is the opposite of $2 + \tfrac{3}{4}$ .
Two wholes are $\frac{8}{4}$ , and
$\frac{8}{4} + \frac{3}{4} = \frac{11}{4}.$
So
$-2\tfrac{3}{4} = \frac{-11}{4}.$

Zero is $\frac{0}{b}$ for every nonzero integer $b$ , because none of the parts are taken.
The smallest positive denominator is $1$ , so in lowest terms
$0 = \frac{0}{1}.$

Answer

$-0.35 = \frac{-7}{20}$ , $-2\tfrac{3}{4} = \frac{-11}{4}$ and $0 = \frac{0}{1}$ . Writing the minus sign in front instead, as $-\tfrac{7}{20}$ and $-\tfrac{11}{4}$ , names the same numbers.

Key idea

Integers, zero and decimals that end are all rational, and the minus sign on a negative mixed number covers the whole number and the fraction together.
Problem 2 Five numbers in a row

Put these numbers in order from least to greatest: $-1.05$ , $-1\tfrac{1}{8}$ , $-0.9$ , $\tfrac{1}{4}$ , $-1$ .
1. Hint 1
  Separate the numbers that lie left of zero from the ones that lie right of it before comparing anything else.
2. Hint 2
  Among negative numbers, the one farther from zero is farther left, so compare their absolute values and remember which way that runs.
3. Hint 3
  Writing $\tfrac{1}{8}$ as the decimal $0.125$ puts all four negative numbers in one form.
Answer

$-1\tfrac{1}{8}$ , $-1.05$ , $-1$ , $-0.9$ , $\tfrac{1}{4}$ .

Full solution

Only $\tfrac{1}{4}$ is positive, so it is the greatest.
The other four numbers are negative, so all of them lie left of zero.

Their absolute values are $1.05$ , $1.125$ , $0.9$ and $1$ , since
$1\tfrac{1}{8} = 1.125.$
In order of size those run
$0.9 < 1 < 1.05 < 1.125.$

On the negative side a larger size is farther from zero, which means farther left, so the order turns around.
The smallest negative is $-1.125$ and the greatest is $-0.9$ .

So from least to greatest:
$-1\tfrac{1}{8} < -1.05 < -1 < -0.9 < \tfrac{1}{4}$

Answer

$-1\tfrac{1}{8}$ , $-1.05$ , $-1$ , $-0.9$ , $\tfrac{1}{4}$ .

Key idea

Between two negative numbers, the one with the larger absolute value is the smaller number.
Problem 3 Three fractions, three signs

Compute $-\tfrac{5}{6} + \tfrac{3}{8} - \left(-\tfrac{1}{4}\right)$ , and give the answer as a fraction in lowest terms.
1. Hint 1
  Subtracting a negative number can be rewritten as adding a positive one, which leaves a sum of three terms.
2. Hint 2
  All three fractions can be written over one denominator, the least common multiple of $6$ , $8$ and $4$ .
3. Hint 3
  Combine the two positive terms first, then set their total against the negative term and ask which side is farther from zero.
Answer

$-\tfrac{5}{24}$

Full solution

Subtracting $-\tfrac{1}{4}$ is adding its opposite, $\tfrac{1}{4}$ .
So the expression is $-\tfrac{5}{6} + \tfrac{3}{8} + \tfrac{1}{4}$ .

The least common multiple of $6$ , $8$ and $4$ is $24$ .
Over $24$ ths the three terms are $-\tfrac{20}{24}$ , $\tfrac{9}{24}$ and $\tfrac{6}{24}$ .

The two positive terms add to $\tfrac{15}{24}$ .
Against $-\tfrac{20}{24}$ the signs differ, so subtract the sizes, and the negative term is farther from zero.
$-\frac{20}{24} + \frac{15}{24} = -\frac{5}{24}$

The numerator $5$ is prime and does not divide $24$ , so $-\tfrac{5}{24}$ is in lowest terms.

Answer

$-\tfrac{5}{24}$

Key idea

Rewrite every subtraction as adding the opposite, then the integer sign rules decide the sign while the fraction methods find the size.
Problem 4 A quotient, then a product

Compute $(-3.6) \div (-0.45) \times \left(-\tfrac{1}{8}\right)$ , working from left to right.
1. Hint 1
  Each step has a sign and a size, and the sign of each step depends only on the signs of its two numbers.
2. Hint 2
  For the division, slide both decimal points the same number of places until the divisor is a whole number.
3. Hint 3
  The quotient turns out to be a whole number, so the last step takes an eighth of it.
Answer

$-1$

Full solution

Start with the division.
Both numbers are negative, so the signs match and the quotient is positive.

Sliding both points two places right makes the divisor whole:
$3.6 \div 0.45 = 360 \div 45.$
Since $45 \times 8 = 360$ , that quotient is $8$ .

Next multiply by $-\tfrac{1}{8}$ .
The signs of $8$ and $-\tfrac{1}{8}$ differ, so the product is negative, and its size is $8 \times \tfrac{1}{8} = 1$ .
$8 \times \left(-\frac{1}{8}\right) = -1$

Counting the negatives gives the same sign.
No number is zero, and three of them are negative, an odd count, so the result is negative.

Answer

$-1$

Key idea

In a chain of multiplications and divisions with no zero, an odd number of negatives gives a negative result and an even number a positive one.
Problem 5 A cold night

At midnight a thermometer reads $-6.4$ degrees. By 3 in the morning the temperature has dropped another $1.8$ degrees. By 9 in the morning it has risen $4.25$ degrees from its 3 o'clock reading.

What does the thermometer read at 9 in the morning? How many degrees below zero was the 3 o'clock reading?
1. Hint 1
  Write every reading as a signed number, then decide whether each change adds or subtracts.
2. Hint 2
  The 3 o'clock reading combines two negative amounts, so its size is a sum of sizes.
3. Hint 3
  How far below zero a reading sits is its distance from zero, which is its absolute value.
Answer

At 9 in the morning the thermometer reads $-3.95$ degrees. The 3 o'clock reading was $-8.2$ degrees, which is $8.2$ degrees below zero.

Full solution

A drop subtracts, so the 3 o'clock reading is $-6.4 - 1.8$ , which is $-6.4 + (-1.8)$ .
Both numbers are negative, so add the sizes and keep the minus sign.
$-6.4 - 1.8 = -8.2$

A rise adds, so the 9 o'clock reading is $-8.2 + 4.25$ .
The signs differ, so subtract the sizes with the points lined up:
$8.20 - 4.25 = 3.95.$
The number $-8.2$ is farther from zero, so the reading is $-3.95$ degrees.

The 3 o'clock reading sits $\lvert -8.2 \rvert = 8.2$ degrees from zero, on the cold side.
So it was $8.2$ degrees below zero.

Answer

At 9 in the morning the thermometer reads $-3.95$ degrees. The 3 o'clock reading was $-8.2$ degrees, which is $8.2$ degrees below zero.

Key idea

A signed number records a reading and its side of zero together, and its absolute value says how far from zero it is.
Problem 6 Paying back part of a loan

Maya owes her brother $7.50$ dollars. She pays back $\tfrac{3}{5}$ of what she owes, and then borrows another $1.25$ dollars from him. Write her balance with her brother as a signed number.
1. Hint 1
  A debt is a negative balance, so paying some of it back moves the balance toward zero.
2. Hint 2
  Work out how many dollars three fifths of the debt is before changing the balance.
3. Hint 3
  Borrowing more money takes the balance further below zero, so it subtracts.
Answer

Her balance is $-4.25$ dollars: she owes her brother $4.25$ dollars.

Full solution

Her starting balance is $-7.5$ dollars.
Three fifths of the $7.5$ dollars she owes is $\tfrac{3}{5} \times 7.5$ .
A fifth of $7.5$ is $1.5$ , and three fifths is $4.5$ , so she pays back $4.5$ dollars.

Paying back adds to the balance.
The signs differ and $-7.5$ is farther from zero.
$-7.5 + 4.5 = -3$

As a check, she still owes two fifths of the debt, and $\tfrac{2}{5} \times (-7.5) = -3$ , a positive times a negative.

Borrowing another $1.25$ dollars subtracts, and two negatives combine.
$-3 - 1.25 = -4.25$

Answer

Her balance is $-4.25$ dollars: she owes her brother $4.25$ dollars.

Key idea

Model a debt as a negative balance: paying back adds, borrowing subtracts, and the integer sign rules do the rest.
Problem 7 An exact sum, and a rounded one

Find the exact value of $-0.6 + \tfrac{5}{6}$ . A classmate writes $\tfrac{5}{6}$ as $0.83$ and gets $0.23$ . Is the classmate's answer exactly right? Explain.
1. Hint 1
  One of the two numbers has a decimal that never ends. Which form of number can hold both of them exactly?
2. Hint 2
  Write $-0.6$ as a fraction, then put both fractions over a common denominator.
3. Hint 3
  To judge the classmate's answer, write $0.23$ as a fraction and compare it with yours over one denominator.
Answer

$-0.6 + \tfrac{5}{6} = \tfrac{7}{30}$ . The classmate's $0.23$ is not exactly right: it is $\tfrac{1}{300}$ less than $\tfrac{7}{30}$ , because $0.83$ is only a rounded value of $\tfrac{5}{6}$ .

Full solution

Dividing $5$ by $6$ gives $0.8333\ldots$ , a repeating decimal with no last digit, so writing out its digits, as in $0.83$ , cuts it short; work with fractions.
The decimal $-0.6$ is
$-\tfrac{6}{10} = -\tfrac{3}{5}.$

Over $30$ ths the two numbers are $-\tfrac{18}{30}$ and $\tfrac{25}{30}$ .
The signs differ and $\tfrac{25}{30}$ is farther from zero, so the sum is positive.
$-\frac{18}{30} + \frac{25}{30} = \frac{7}{30}$

The classmate's $0.23$ is $\tfrac{23}{100}$ .
Over $300$ ths, $\tfrac{23}{100} = \tfrac{69}{300}$ and $\tfrac{7}{30} = \tfrac{70}{300}$ , so the two differ by $\tfrac{1}{300}$ .

The gap came from rounding $\tfrac{5}{6}$ to $0.83$ before adding.
The answer $0.23$ is close to the exact value, but it is not equal to it.

Answer

$-0.6 + \tfrac{5}{6} = \tfrac{7}{30}$ . The classmate's $0.23$ is not exactly right: it is $\tfrac{1}{300}$ less than $\tfrac{7}{30}$ , because $0.83$ is only a rounded value of $\tfrac{5}{6}$ .

Key idea

When a fraction's decimal repeats, compute with the fraction; a rounded decimal gives an answer that is close but not exact.
Problem 8 Where the minus signs sit

Decide which of these five numbers are equal to $-\tfrac{4}{9}$ , and explain each decision: $\frac{-4}{-9}$ , $\frac{4}{-9}$ , $-\left(\frac{-4}{9}\right)$ , $\frac{-8}{18}$ and $-0.\overline{4}$ .
1. Hint 1
  Read each fraction bar as a division and decide its sign from the signs of the top and the bottom.
2. Hint 2
  A minus sign in front of parentheses asks for the opposite of whatever is inside them.
3. Hint 3
  For the repeating decimal, divide $4$ by $9$ and look at which digit keeps coming back.
Answer

$\frac{4}{-9}$ , $\frac{-8}{18}$ and $-0.\overline{4}$ are equal to $-\tfrac{4}{9}$ . The other two, $\frac{-4}{-9}$ and $-\left(\frac{-4}{9}\right)$ , are both equal to $\tfrac{4}{9}$ .

Full solution

In $\frac{-4}{-9}$ the top and bottom are both negative, so the signs match and the quotient is positive.
It equals $\tfrac{4}{9}$ , not $-\tfrac{4}{9}$ .

In $\frac{4}{-9}$ the signs differ, so the quotient is negative with size $\tfrac{4}{9}$ .
It equals $-\tfrac{4}{9}$ .

Inside the parentheses, $\frac{-4}{9}$ is $-\tfrac{4}{9}$ .
The minus sign in front takes its opposite, which is $\tfrac{4}{9}$ , so this one is not equal.

The fraction $\frac{-8}{18}$ is negative, and dividing $8$ and $18$ by $2$ gives $\frac{-4}{9}$ .
So it is equal:
$\frac{-8}{18} = \frac{-4}{9} = -\frac{4}{9}$

Dividing $4$ by $9$ leaves a remainder of $4$ at every step, so every digit is $4$ and
$\tfrac{4}{9} = 0.\overline{4}.$
Its opposite, $-0.\overline{4}$ , is therefore $-\tfrac{4}{9}$ .

Answer

$\frac{4}{-9}$ , $\frac{-8}{18}$ and $-0.\overline{4}$ are equal to $-\tfrac{4}{9}$ . The other two, $\frac{-4}{-9}$ and $-\left(\frac{-4}{9}\right)$ , are both equal to $\tfrac{4}{9}$ .

Key idea

A fraction with a nonzero numerator is negative exactly when its top and bottom have different signs, and a minus sign in front then gives its opposite.
Problem 9 Checking Jordan's product

Jordan computes $\left(-2\tfrac{1}{3}\right) \times 1\tfrac{1}{2}$ like this: $-2\tfrac{1}{3} = -2 + \tfrac{1}{3} = -\tfrac{5}{3}$ , so the product is $-\tfrac{5}{3} \times \tfrac{3}{2} = -\tfrac{5}{2}$ . Find Jordan's mistake, and give the correct product as a mixed number.
1. Hint 1
  Check Jordan's first step on the number line. Where does $-2\tfrac{1}{3}$ sit compared with $-2$ , and where does $-\tfrac{5}{3}$ sit?
2. Hint 2
  The minus sign covers the whole mixed number, so $-2\tfrac{1}{3}$ is the opposite of $2 + \tfrac{1}{3}$ .
3. Hint 3
  Once both factors are fractions, settle the sign of the product, then multiply the tops and the bottoms.
Answer

Jordan's first step is wrong: $-2\tfrac{1}{3}$ is $-\tfrac{7}{3}$ , not $-2 + \tfrac{1}{3}$ . The correct product is $-3\tfrac{1}{2}$ .

Full solution

The number $-2\tfrac{1}{3}$ lies a third of a unit to the left of $-2$ .
But $-2 + \tfrac{1}{3} = -1\tfrac{2}{3}$ lies to the right of $-2$ , so Jordan's first step changed the number.

The minus sign belongs to the whole mixed number.
$-2\tfrac{1}{3} = -\left(\frac{6}{3} + \frac{1}{3}\right) = -\frac{7}{3}$

One factor is negative and one is positive, so the product is negative.
For the size, multiply the tops and the bottoms.
$\frac{7}{3} \times \frac{3}{2} = \frac{7}{2}$

So the product is $-\tfrac{7}{2}$ , which is $-3\tfrac{1}{2}$ as a mixed number.
As a check, $2\tfrac{1}{3}$ times one and a half should be a little more than $2 \times 1.5 = 3$ , and $3\tfrac{1}{2}$ is.

Answer

Jordan's first step is wrong: $-2\tfrac{1}{3}$ is $-\tfrac{7}{3}$ , not $-2 + \tfrac{1}{3}$ . The correct product is $-3\tfrac{1}{2}$ .

Key idea

A negative mixed number is the opposite of the whole mixed number, so its minus sign applies to the fraction part as well as the whole part.
Problem 10 Two divisions that use the same parentheses

Decide whether $(-1.5) \div \left(\tfrac{2}{3} - 0.75 + \tfrac{1}{12}\right)$ has a value, and explain. Then do the same for $\left(\tfrac{2}{3} - 0.75 + \tfrac{1}{12}\right) \div (-1.5)$ , and give its value if it has one.
1. Hint 1
  Work out the parentheses before dividing. The value inside decides what happens in both divisions.
2. Hint 2
  Write $0.75$ as a fraction so that all three terms in the parentheses share the denominator $12$ .
3. Hint 3
  For each division, ask what number times the divisor would give back the number being divided.
Answer

The parentheses equal $0$ . So $(-1.5) \div \left(\tfrac{2}{3} - 0.75 + \tfrac{1}{12}\right)$ is undefined, while $\left(\tfrac{2}{3} - 0.75 + \tfrac{1}{12}\right) \div (-1.5) = 0$ .

Full solution

Inside the parentheses,
$0.75 = \tfrac{3}{4}.$
Over twelfths the three terms are $\tfrac{8}{12}$ , $-\tfrac{9}{12}$ and $\tfrac{1}{12}$ .
$\frac{8}{12} - \frac{9}{12} + \frac{1}{12} = 0$

So the first expression is $(-1.5) \div 0$ .
That would be a number that times $0$ gives $-1.5$ , but every number times $0$ is $0$ .
No such number exists, so the first expression has no value: it is undefined.

The second expression is $0 \div (-1.5)$ .
A number that times $-1.5$ gives $0$ does exist, and there is only one, the number $0$ .
So the second expression equals $0$ .

Answer

The parentheses equal $0$ . So $(-1.5) \div \left(\tfrac{2}{3} - 0.75 + \tfrac{1}{12}\right)$ is undefined, while $\left(\tfrac{2}{3} - 0.75 + \tfrac{1}{12}\right) \div (-1.5) = 0$ .

Key idea

Zero divided by a nonzero number is zero, but no number can be divided by zero, whatever form that zero arrives in.

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