The Lumenary · Grades 3–5

Fractions and decimals

Name: ______________________

Date: ______________________

You will learn to

Place fractions on a number line and explain $\tfrac{a}{b}$ as $a$ copies of $\tfrac{1}{b}$
Make equivalent fractions, and test whether two fractions are equal by cross-multiplying
Compare, add and subtract fractions using a common denominator, and explain why $\tfrac12 + \tfrac13$ is not $\tfrac25$
Convert between decimals with up to three places and fractions with denominators $10$ , $100$ and $1000$
Multiply a fraction by a whole number

1.
Fill in the missing numerator: $\dfrac{3}{4} = \dfrac{?}{20}$ .
2.
On a number line, which point is further from $0$ : $\tfrac18$ or $\tfrac15$ ?
- A. $\tfrac15$
- B. $\tfrac18$
- C.They are the same distance from $0$
3.
Sipho says $\tfrac12 + \tfrac13 = \tfrac25$ . What is the best reply?
- A.That can't be right: $\tfrac25$ is less than $\tfrac12$ , but the sum must be more than $\tfrac12$ . The sum is $\tfrac56$ .
- B.Right: add the tops ( $1 + 1 = 2$ ) and the bottoms ( $2 + 3 = 5$ ).
- C.Nearly: the answer is $\tfrac26$ .
- D.The answer is $\tfrac56$ , but only because that is the rule.
4.
Work out $\dfrac{2}{3} + \dfrac{1}{4}$ . Write the answer in twelfths: $\dfrac{?}{12}$ .
5.
Kiri ran $0.25$ km and Ana ran $0.3$ km. Who ran further?
- A.Ana: $0.3$ km is $30$ hundredths, more than $25$ hundredths
- B.Kiri, because $25$ is bigger than $3$
- C.They ran the same distance
6.
Write $0.125$ as a fraction with denominator $1000$ : $0.125 = \dfrac{?}{1000}$ .
7.
What is $4 \times \tfrac23$ ?
- A. $\tfrac83$ , which is $2\tfrac23$
- B. $\tfrac{8}{12}$
- C. $\tfrac63$
- D. $\tfrac{2}{12}$
8.
Which of these fractions have a decimal that stops? (Choose all that apply.)
- A. $\tfrac38$
- B. $\tfrac13$
- C. $\tfrac{7}{20}$
- D. $\tfrac16$
- E. $\tfrac{3}{12}$
9.
In Lean, Gideon says that the fractions $\tfrac{a}{b}$ and $\tfrac{c}{d}$ are equivalent when a * d = c * b. Which statement does Lean need to check to show that $\tfrac24$ is equivalent to $\tfrac12$ ?
- A.2 * 2 = 1 * 4
- B.2 * 1 = 4 * 2
- C.2 + 2 = 1 + 4
- D.2 * 4 = 1 * 2
10.
Write add_fractions(a, b, c, d) that returns the sum $\tfrac{a}{b} + \tfrac{c}{d}$ as a pair (numerator, denominator) in simplest form. The starter uses the wrong rule from the lesson, so the tests fail. Use $\tfrac{a \times d + c \times b}{b \times d}$ , then divide top and bottom by their greatest common factor, math.gcd.
Computational — Practice

Answer key — Fractions and decimals

1.
$4 \times 5 = 20$ , so we cut every quarter into $5$ pieces: $\tfrac34 = \tfrac{3 \times 5}{4 \times 5} = \tfrac{15}{20}$ . Check with cross-multiplication: $3 \times 20 = 60 = 15 \times 4$ .
2.
$\tfrac15$ is further from $0$ . The whole is cut into fewer parts, so each part is longer. With the cross-multiplication test, $1 \times 5 = 5 < 8 = 1 \times 8$ , so $\tfrac18 < \tfrac15$ .
3.
$\tfrac25 < \tfrac12$ (since $2 \times 2 = 4 < 5 = 1 \times 5$ ), but $\tfrac12 + \tfrac13$ must be bigger than $\tfrac12$ . So $\tfrac25$ is impossible. Correctly: $\tfrac12 + \tfrac13 = \tfrac36 + \tfrac26 = \tfrac56$ .
4.
$\tfrac23 = \tfrac{8}{12}$ and $\tfrac14 = \tfrac{3}{12}$ . So $\tfrac23 + \tfrac14 = \tfrac{8}{12} + \tfrac{3}{12} = \tfrac{11}{12}$ .
5.
$0.3 = 0.30 = \tfrac{30}{100}$ and $0.25 = \tfrac{25}{100}$ . Since $30 > 25$ , Ana ran further.
6.
$0.125 = \tfrac{100}{1000} + \tfrac{20}{1000} + \tfrac{5}{1000} = \tfrac{125}{1000}$ . And since $8 \times 125 = 1000$ , this is also $\tfrac18$ .
7.
$4 \times \tfrac23 = \tfrac23 + \tfrac23 + \tfrac23 + \tfrac23 = \tfrac83$ . That is $2$ wholes ( $\tfrac63$ ) and $\tfrac23$ more: $2\tfrac23$ . In general $n \times \tfrac{a}{b} = \tfrac{n \times a}{b}$ .
8.
$\tfrac38 = 0.375$ , $\tfrac{7}{20} = 0.35$ and $\tfrac{3}{12} = \tfrac14 = 0.25$ stop. $\tfrac13 = 0.333\ldots$ and $\tfrac16 = 0.1666\ldots$ go on forever. In lowest terms, a decimal stops exactly when the denominator is made only of $2$ s and $5$ s.
9.
With $a = 2$ , $b = 4$ , $c = 1$ , $d = 2$ , the test a * d = c * b becomes 2 * 2 = 1 * 4, which is $4 = 4$ . True, so the fractions are equivalent. This is the first part of the Lean theorem two_quarters.
10.
```
import math

def add_fractions(a, b, c, d):
    top = a * d + c * b
    bottom = b * d
    g = math.gcd(top, bottom)
    return (top // g, bottom // g)
```
For $\tfrac14 + \tfrac14$ : the rule gives $\tfrac{4 + 4}{16} = \tfrac{8}{16}$ , and dividing by $\gcd(8, 16) = 8$ gives $\tfrac12$ . The tests try a few cases. The lesson's argument (same-sized pieces, then count) shows the rule is right for every pair of fractions.