SHSAT: Signed numbers and fractions: track every sign
Combine signed changes, separate net change from distance, and avoid common sign and fraction slips.
How do you keep signs straight when a problem mixes rises and falls, fractions and powers?
Short answer
Start from the starting value and write each change as a signed number. Keep fractions exact, and apply exponents before a leading minus sign.
Know first: Adding fractions with a common denominator, Absolute value
Step 1 of 5
Give every movement a sign
Signed-number questions describe movement in words: rises, falls, deposits, moves left. Turn each word into a positive or negative number before you add anything.
Two different questions hide in these stories. The net change is the signed total, where ups and downs cancel. The distance traveled adds the sizes of the moves, so nothing cancels.
| You read | You write | Note |
|---|---|---|
| rises, right, gains, deposit | + | a positive change |
| falls, left, loses, withdrawal | − | a negative change |
| change from A to B | B − A | final minus starting value |
| total distance | |move 1| + |move 2| + … | add sizes, ignore signs |
Write the starting value first, then each signed change.
Step 2 of 5
Track position and distance separately
Worked example: A bug on a number line
A bug starts at −3/4 on a number line. It moves right 5/2 units, then left 7/4 units. Where does it finish?
- A−5
- B0
- C3/4
- D17/4
- 1
Use one denominator
Fourths: start at −3/4, right is +10/4, left is −7/4.
- 2
Add the signed moves
−3/4 + 10/4 − 7/4 = 0/4 = 0.
- 3
Keep distance apart
The bug traveled 10/4 + 7/4 = 17/4 units, but it finished at 0. Distance and position answer different questions.
- 4
See why the other choices are there
−5 treats both moves as left. 3/4 adds the moves but forgets the starting point. 17/4 is the distance traveled, not the position.
Answer
The bug finishes at 0.
Find the wrong step
The temperature falls from −1.8°C to −6.3°C. What is the signed change?
- 1
Change is final minus starting value.
- 2
−6.3 − (−1.8) = −6.3 − 1.8
What went wrong
Subtracting a negative number is the same as adding its opposite. The two minus signs make a plus.
- 3
So the change is −8.1°C.
The fix
−6.3 − (−1.8) = −6.3 + 1.8 = −4.5°C. The answer is negative because the temperature fell.
Step 3 of 5
Signs, fractions and complete pieces
Check yourself · Question 1
What is the value of (2/3 − 5/6) ÷ (−1/4)?
Answer: C
In sixths, 4/6 − 5/6 = −1/6. Dividing by −1/4 means multiplying by −4: (−1/6) × (−4) = 4/6 = 2/3. Two negatives give a positive.
Trap answer: 1/24
Multiplying by the divisor
- Why it looks right
- The fractions are small, and multiplying them is the most familiar move.
- Why it's wrong
- Division by −1/4 asks how many quarters fit, which makes the result larger in size. Multiplying by −1/4 makes it smaller.
- The right answer
- 2/3: flip the divisor and multiply, (−1/6) × (−4) = 2/3.
Check yourself · Question 2
A rope is 7 1/2 meters long. Each jump rope needs 2/3 meter. How many complete jump ropes can be cut from it?
Answer: B
7 1/2 ÷ 2/3 = 15/2 × 3/2 = 45/4 = 11 1/4. Only 11 whole pieces fit, because the leftover 1/4 of a piece is too short.
Check yourself · Question 3
A balloon changes height by −6 meters, then +10 meters, then −2 meters. How much greater is the total distance it moved than the size of its net change?
Answer: B
Distance: 6 + 10 + 2 = 18 meters. Net change: −6 + 10 − 2 = 2, whose size is 2. The difference is 18 − 2 = 16.
Common mistake
−6² is 36, because a negative times a negative is positive.
- Why it's tempting
- You remember that (−6)(−6) = 36, and the two expressions look almost the same.
- Do this instead
- Without parentheses the exponent applies only to 6, so −6² = −36. Write the parentheses yourself whenever the base is negative.
Step 4 of 5
What to remember
Remember
- Turn each movement word into a signed number, and find change as final minus starting value.
- Net change lets moves cancel; total distance adds their sizes.
- Power before the minus sign, flip and multiply to divide fractions, and round down when only complete pieces count.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, what signed number did you write for each movement, and what was your starting value?