Cryptic Combinations

Solution

Page 1: Jackson Page 2: Isabella Page 3: Sebastian
Left 42
Right 37
Left 20
Left 25
Right 49
Left 9
Right 16
Left 43
Right 23
Left 33

Explanation

First, follow the order of operations to find the numbers in each combination.

The clues for the combination on page 1:

Turn left: (2² − 10) (2.6 + -9.6)

Evaluate powers within parentheses.
Complete operations within parentheses.
(2² − 10) (2.6 + -9.6)
(4 − 10) (2.6 + -9.6)
(-6) (-7)
Multiply and divide in order from left to right. (-6) (-7)
42

Turn right: negative one-third (-46.3 − 28.7) + twenty-four-eighths x sixteen-fourths

Complete operations within parentheses. negative one-third (-46.3 − 28.7) + twenty-four-eighths x sixteen-fourths
negative one-third (-75) + twenty-four-eighths x sixteen-fourths
Multiply and divide in order from left to right. 25 + twenty-four-eighths x sixteen-fourths
25 + 3 x 4
25 + 12
Add and subtract in order from left to right. 25 + 12
37

Turn left: (9² − 1) x (negative one-half)²

Evaluate powers within parentheses. Complete operations within parentheses. (9² − 1) x (negative one-half)²
(81 − 1) x (negative one-half)²
(80) x (negative one-half)²
Evaluate powers. (80) x (negative one-half)²
(80) x one-fourth
Multiply and divide in order from left to right. (80) x one-fourth
20

The combination for page 1 is:

Left 42
Right 37
Left 20

42 + 37 + 20 = 99

The clues for the combination on page 2:

Turn left: one-fourth (27.8 + 8.2) + one-third (-6 x -8)

Complete operations within parentheses. one-fourth (27.8 + 8.2) + one-third (-6 x -8)
one-fourth (36) + one-third (48)
Multiply and divide in order from left to right. one-fourth (36) + one-third (48)
9 + one-third (48)
Multiply and divide in order from left to right. 9 + one-third (48)
9 + 16
Add and subtract in order from left to right. 9 + 16
25

Turn right: (-8 + 2 − -10 + 3) (2² + 3 x 12)

Evaluate powers within parentheses. First, complete operations within parentheses. Next, multiply and divide.

Last, add and subtract from left to right.

(-8 + 2 − -10 + 3) (2² + 3 x 12)
(-8 + 2 − -10 + 3) (4 + 3 x 12)
(-8 + 2 − -10 + 3) (4 + 3 x 12)
(-8 + 2 − -10 + 3) (4 + 3 x 12)
(-8 + 2 − -10 + 3) (4 + 3)
(-6 + 10 + 3) (4 + 3)
(+4 + 3) (4 + 3)
(7) (7)
Multiply and divide in order from left to right. (7) (7)
49

Turn left: -8 (-9 +7)² ÷ (-19.8 + 17.8) − 7

Complete operations within parentheses. -8 (-9 + 7)² ÷ (-19.8 + 17.8) − 7
-8 (-2)² ÷ (-2) − 7
Evaluate powers. -8 (-2)² ÷ (-2) − 7
-8 (4) ÷ (-2) − 7
Multiply and divide in order from left to right. -32 ÷ -2 − 7
16 − 7
Add and subtract in order from left to right. 16 − 7
9

Turn right: (15 ÷ three-fifths ) − (3.4² − 2.56)

Evaluate powers within parentheses. (15 ÷ three-fifths ) − (3.4² − 2.56)
(15 ÷ three-fifths ) − (11.56 − 2.56)
Complete operations within parentheses. (15 ÷ three-fifths ) − (11.56 − 2.56)
(15 x five-thirds) − (11.56 − 2.56)
(seventy-five over three) − (11.56 − 2.56)
(25) − (9)
Add and subtract in order from left to right. (25) − (9)
16

The combination for page 2 is:

Left 25
Right 49
Left 9
Right 16

25 + 49 + 9 + 16 = 99

The clues for the combination on page 3:

Turn left: (-12 + 4)² − 3 (6 + 1)

Complete operations within parentheses. (-12 + 4) ² − 3 (6 + 1)
(-8)² − 3 (7)
Evaluate powers. (-8)² − 3 (7)
64 − 3 (7)
Multiply and divide in order from left to right. 64 − 3 (7)
64 − 21
Add and subtract in order from left to right. 64 − 21
43

Turn right: (3 one-half ÷ 5 one-fourth ) (-12 x -5) − (4² + 1)

Complete operations within parentheses. (3 one-half ÷ 5 one-fourth ) (-12 x -5) − (4² + 1)
(seven-halves x four over 21 ) (+60) − (16 + 1)
(two-thirds) (60) − 17
Multiply and divide in order from left to right. (two-thirds) (60) − 17
40 − 17
Add and subtract in order from left to right. 40 − 17
23

Turn left: -4 (-3 − 4.5) + (1.25 + 1.75)

Complete operations within parentheses. -4 (-3 − 4.5) + (1.25 + 1.75)
-4 (-7.5) + (3)
Multiply and divide in order from left to right. -4 (-7.5) + (3)
30 + (3)
Add and subtract in order from left to right. 33

The combination for page 3 is:

Left 43
Right 23
Left 33

43 + 23 + 33 = 99

Summary

Match the clues to the owner:

  • All the numbers in Isabella's combination are perfect squares. All the numbers on page 2 are perfect squares. Page 2 is Isabella's combination.
  • Only 1 number in Jackson's lock combination is odd. Two of the numbers on Page 1 are even and only one of them is odd. (All the numbers on Page 3 are odd.) Page 1 is the combination to Jackson's lock.
  • The remaining combination,on Page 3, must belong to Sebastian.

To check the combinations, add the totals:

Page 1 Page 2 Page 3
Left 42
Right 37
Left 20
Left 25
Right 49
Left 9
Right 16
Left 43
Right 23
Left 33
Total 99 Total 99 Total 99


Houghton Mifflin Math Grade 6