Depth · one concept from first principles · step 2 of 4
Split by common divisors, count |b| values
Count integer pairs (a, b) where a divides both 8 and 12, and |b|/a is at least 1/2 but less than 2.
Number system
Your answer
How to solve
Strategy Make a Systematic List — Condition (a) leaves only a few choices for a, so split into one case per a. In each case, condition (b) becomes a range for |b|, and each value of |b| gives two values of b, one positive and one negative.
1STEP 1
Find the sign of a
In (b) the ratio |b|/a is at least 1/2, so it is positive. Its top |b| is never negative, so the bottom a must be positive (and b ≠ 0).
|b|/a ≥ 1/2 ＞ 0, |b| ≥ 0 → a ＞ 0
A quotient is positive only when top and bottom have the same sign.
7.NS.A.2Identify Subproblems2STEP 2
List the choices for a
By (a), a is a positive common divisor of 8 and 12, that is, a divisor of 4.
a ∈ {1, 2, 4}
Only three cases: a = 1, 2, 4.
6.NS.B.4Identify Subproblems3STEP 3
Turn (b) into a range for |b|
Multiply each part of (b) by the positive number a.
a/2 ≤ |b| ＜ 2a
Multiplying by a positive number keeps the inequality directions.
6.NS.C.7Make A Systematic ListBecause is positive, condition (b) says exactly .
Why?
must be positive, because the ratio is at least while its top is never negative.
Why?
is how far sits from , so it is never negative.
🧱Absolute value is distance from zero|a| is how far a sits from 0 on the number line, whichever side it is on, so it is never negative and |a - b| is the gap between a and b.
Why?
A quotient with unlike signs is negative, so a positive quotient whose top is not negative needs a positive bottom.
🧱Signs of products and quotientsMultiplying by a negative number flips a number to the other side of zero, so two flips land back on the positive side: for two nonzero numbers, a product or quotient with like signs is positive and one with unlike signs is negative (zero itself has no sign).
Why?
Multiplying every part of (b) by the positive number keeps each inequality pointing the same way.
🧱Scaling an inequalityMultiplying both sides by the same positive number stretches or shrinks the gaps on the number line but keeps the order; multiplying by the same negative number also flips the line around zero, so the inequality turns over.
4STEP 4
Read off |b| in each case
a = 1: 1/2 ≤ |b| ＜ 2. a = 2: 1 ≤ |b| ＜ 4. a = 4: 2 ≤ |b| ＜ 8.
a=1: |b| = 1; a=2: |b| = 1, 2, 3; a=4: |b| = 2, 3, …, 7
That is 1, 3 and 6 values of |b|.
6.NS.C.7Make A Systematic List5STEP 5
Count the pairs
Each value of |b| gives b and -b, so double each count and add.
2 × 1 + 2 × 3 + 2 × 6 = 2 + 6 + 12 = 20
2 + 6 + 12 = 20 pairs.
6.NS.C.7Make A Systematic ListAnswer
20
Spot-check a pair from each case: (1, -1) gives 1/1 = 1; (2, 3) gives 3/2; (4, -7) gives 7/4, all in [1/2, 2). And (4, 8) gives exactly 2, correctly excluded.
Takeaway
Pin down the few possible denominators first, then each case is a simple count.
- Find the sign of a
- List the choices for a
- Turn (b) into a range for |b|
- Read off |b| in each case
- Count the pairs
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested
A parent dashboard for the family lives at sensimlab.com.