Multiplicative Comparison and Factors: practice solutions, Grade 4 – download the PDF

1 Reading a multiplication fact ★★★

Since \( 4 \times 7 = 28 \), the product can be compared with either factor.

“28 is 4 times as many as 7” and “28 is 7 times as many as 4.”

2 Puppy and dog ★★★

Let \( d \) be the weight of the dog in pounds. Then \( d = 4 \times 8 \).

\( 4 \times 8 = 32 \). The dog weighs 32 pounds.

3 Tall tree ★★★

Multiply the height of the bush by 7: \( 7 \times 5 = 35 \).

The oak tree is 35 feet tall.

4 Factor pairs of 18 ★★★

Test 1, 2, 3: \( 1 \times 18 = 18 \), \( 2 \times 9 = 18 \), \( 3 \times 6 = 18 \). Four does not work (\( 4 \times 4 = 16 \), \( 4 \times 5 = 20 \)), and the next test, 5, fails too, so we stop.

Factor pairs: 1 and 18, 2 and 9, 3 and 6. Factors: 1, 2, 3, 6, 9, 18.

5 Skip counting by 8 ★★★

Skip count by 8: 8, 16, 24, 32, 40, 48. Check the last one: \( 6 \times 8 = 48 \).

6 True or false? ★★★

1. True: \( 6 \times 7 = 42 \).
2. False: the multiples of 4 are 4, 8, …, 24, 28, … and 25 is skipped.
3. True: \( 6 \times 9 = 54 \).
4. False: 1 has only one factor. A prime number needs exactly two.

7 Ones digit test ★★★

Look at the ones digit.

• 340: ones digit 0, so divisible by 2, 5 and 10.
• 215: ones digit 5, so divisible by 5 only.
• 462: ones digit 2, so divisible by 2 only.
• 1,000: ones digit 0, so divisible by 2, 5 and 10.

8 More or times as many? ★★★

Omar: \( 3 \times 6 = 18 \) pages. Pia: \( 6 + 3 = 9 \) pages.

\( 18 - 9 = 9 \). Omar read 18 pages and Pia read 9 pages, so Omar read 9 more pages than Pia.

9 Trail length ★★★

Let \( n \) be the number of times. \( 56 = n \times 8 \), so \( n = 56 \div 8 = 7 \).

The bike path is 7 times as long as the walking trail.

10 Factors of 48 ★★★

\( 1 \times 48 \), \( 2 \times 24 \), \( 3 \times 16 \), \( 4 \times 12 \), \( 6 \times 8 \). Five and seven are not factors. After 6 and 8 the pairs would repeat, so we stop.

The factors are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48. So 48 has 10 factors.

11 Sorting numbers ★★★

• 23: prime (not divisible by 2, 3 or 5, as \( 3 \times 7 = 21 \) and \( 3 \times 8 = 24 \)).
• 39: composite, \( 3 \times 13 = 39 \).
• 51: composite, \( 3 \times 17 = 51 \).
• 59: prime (not divisible by 2, 3, 5, or 7, as \( 7 \times 8 = 56 \) and \( 7 \times 9 = 63 \)).
• 77: composite, \( 7 \times 11 = 77 \).
• 91: composite, \( 7 \times 13 = 91 \).

12 Digit sums ★★★

• 5,274: \( 5 + 2 + 7 + 4 = 18 \), so divisible by 3 and by 9.
• 3,185: \( 3 + 1 + 8 + 5 = 17 \), so by neither.
• 7,326: \( 7 + 3 + 2 + 6 = 18 \), so by 3 and by 9.
• 4,512: \( 4 + 5 + 1 + 2 = 12 \), so by 3 only, since 12 is not a multiple of 9.

13 Add 5 pattern ★★★

7, 12, 17, 22, 27, 32, 37, 42.

The ones digits alternate between 7 and 2. The terms alternate between odd and even.

14 Tile figures ★★★

Each figure has 4 tiles in each column, and Figure \( n \) has \( n \) columns, so it has \( 4 \times n \) tiles.

1. \( 4 \times 7 = 28 \) tiles.
2. \( 40 \div 4 = 10 \): Figure 10.
3. No. The number of tiles is a multiple of 4, and 50 is not: \( 4 \times 12 = 48 \) and \( 4 \times 13 = 52 \).

15 Markers for two schools ★★★

First school: \( 9 \times 6 = 54 \) markers.

Let \( m \) be the markers of the second school: \( m = 3 \times 54 = 162 \).

The second school bought 162 markers.

16 Mystery prime ★★★

The primes between 50 and 60 are 53 and 59, because 51, 55 and 57 are divisible by 3 or 5, and the even numbers are composite.

Digit sums: \( 5 + 3 = 8 \) and \( 5 + 9 = 14 \). I am 59.

17 Mystery multiple ★★★

The multiples of 24 are the multiples of both 6 and 8 in this range: 48, 72, 96. (Check: \( 6 \times 8 = 48 \), \( 6 \times 12 = 72 \), \( 6 \times 16 = 96 \), and 8 divides each of them.)

Digit sums: \( 4 + 8 = 12 \), \( 7 + 2 = 9 \), \( 9 + 6 = 15 \). I am 72.

18 Most factors ★★★

List the factors: 21: 1, 3, 7, 21 (4). 22: 1, 2, 11, 22 (4). 23: 1, 23 (2). 24: 1, 2, 3, 4, 6, 8, 12, 24 (8). 25: 1, 5, 25 (3). 26: 1, 2, 13, 26 (4). 27: 1, 3, 9, 27 (4). 28: 1, 2, 4, 7, 14, 28 (6). 29: 1, 29 (2).

The number 24 has the most factors: 8 factors.

19 True or false: prove it ★★★

1. False. The number 2 is prime and even.
2. True. For example, 18 is divisible by 9 and by 3. It works because 9 is itself a multiple of 3 (\( 9 = 3 \times 3 \)).
3. False. The number 5 is a multiple of 5 (\( 5 = 1 \times 5 \)) and is prime. Every other multiple of 5 is composite.

20 Subtract 7 pattern ★★★

1. 100, 93, 86, 79, 72, 65, 58, 51, 44.
2. The first term less than 50 is 44, the ninth term, since 51 is still greater than 50.
3. Yes for 30: \( 100 - 70 = 30 \) and \( 70 = 10 \times 7 \), so it is the 11th term. No for 35: \( 100 - 35 = 65 \), and 65 is not a multiple of 7 (\( 7 \times 9 = 63 \), \( 7 \times 10 = 70 \)).

21 Chairs in rows ★★★

The factor pairs of 36 are 1 and 36, 2 and 18, 3 and 12, 4 and 9, 6 and 6. We cannot use 1 and 36.

She can make 4 rectangles: 2 rows of 18, 3 rows of 12, 4 rows of 9, or 6 rows of 6.

22 Movie and cartoon ★★★

1. \( 90 = n \times 15 \), so \( n = 90 \div 15 = 6 \). The movie is 6 times as long.
2. \( 90 - 15 = 75 \). The movie is 75 minutes longer.
3. Part (a) compares by multiplication (how many copies of 15 fit in 90), and part (b) compares by subtraction (the difference between the lengths). They answer two different questions.