SME
Grade 9 Math · Problem type
Tolerance: within d of a target
Use this for “A 350 g box may be off by 6 g. Is a 343 g box all right?”
Diagram
|w − 350| ≤ 6: every number at most 6 from 350
350 − 6 = 344
350 + 6 = 356
So 344 ≤ w ≤ 356.
Both circles are closed: 344 and 356 are taken in.
Test 343 g: |343 − 350| = 7, and 7 ≤ 6 is false.
This is an example. Type your own values to start a new problem.
|
w (g)
−
T (g)
|
≤
d (g)
L (g)
≤
w
≤
U (g)
k
Distance from the target
worked out (not typed) · g
h
Is the test true?
worked out (not typed)
Formulas
L
=
T
−
d
344 = 350 − 6
U
=
T
+
d
356 = 350 + 6
k
=
|
w
−
T
|
7 = |343 − 350|
test:
k
≤
d
gives
h
test: 7 ≤ 6 gives false
Assumptions
•
Within d of the target T means |w − T| ≤ d: the weight is at most d from T.
•
So T − d ≤ w ≤ T + d, both ends allowed.
•
A weight is all right when its distance from T is at most d.
Step-by-step
WE KNOW
T = 350 g, d = 6 g, w = 343 g
FIND
lowest allowed weight (L), highest allowed weight (U), distance from the target (k), is the test true? (h)
Step 1 · Find lowest allowed weight (L)
Use L = T − d
The
lightest
box
allowed:
d
below
the
target.
L
=
350
−
6
L
=
344
g
Step 2 · Find highest allowed weight (U)
Use U = T + d
The
heaviest
box
allowed:
d
above
the
target.
U
=
350
+
6
U
=
356
g
Step 3 · Find distance from the target (k)
Use k = |w − T|
How far the weight is from the target, as a distance (never negative).
k
=
|343
−
350|
k
=
|−7|
k
=
7
g
Step 4 · Find is the test true? (h)
Use test: k ≤ d gives h
Compare the distance with the allowed difference: true means it is all right.
7 ≤ 6 is false
h
=
false
Check
Put the numbers back in. Both sides should match.
344 = 350 − 6 ✓
356 = 350 + 6 ✓
7 = |343 − 350| ✓
7 > 6, so the weight is not within the allowed difference ✓
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