SME
Grade 8 Math · Problem type
Which is bigger, and how many times
Use this for “Which is greater, 2.4 × 10⁷ or 6.5 × 10⁴, and about how many times?”
Diagram
1,000,000 = 1 × 10⁶
10⁶ ≤ 1,000,000 < 10⁷
10⁶ = 1,000,000
10⁷ = 10,000,000
t = 5 × 10² (first ÷ second)
T = 5 × 10² (times as big)
This is an example. Type your own values to start a new problem.
(
a
×
10
)
÷
(
c
×
10
)
=
t
T
Times as big
worked out (not typed)
Formulas
t
=
(
a
÷
c
)
×
10^(
n
−
k
)
(first
÷
second
=
(front
of
the
first
÷
front
of
the
second)
×
10^(power
of
the
first
−
power
of
the
second))
5 ×
10
2
=
(1
÷
2)
×
10^(6
−
3)
T
=
t
,
or
1
÷
t
when
t
<
1
(times
as
big
=
first
÷
second,
or
1
÷
first
÷
second
when
first
÷
second
<
1)
5 ×
10
2
=
5 ×
10
2
,
or
1
÷
(5
×
10
2
)
when
5 ×
10
2
<
1
Assumptions
•
Compare the powers of ten first: the bigger exponent wins.
•
With the same power, compare the numbers in front.
•
To find how many times, divide the fronts and subtract the exponents.
Step-by-step
WE KNOW
a = 1 (front of the first), n = 6 (power of the first), c = 2 (front of the second), k = 3 (power of the second)
FIND
first ÷ second (t), times as big (T)
Step 1 · Find first ÷ second (t)
Use t = (a ÷ c) × 10ⁿ⁻ᵏ (first ÷ second = (front of the first ÷ front of the second) × 10ᵖᵒʷᵉʳᵒᶠᵗʰᵉᶠⁱʳˢᵗ⁻ᵖᵒʷᵉʳᵒᶠᵗʰᵉˢᵉᶜᵒⁿᵈ)
Divide the fronts, and subtract the exponents for the powers of ten.
t
=
(
1
2
)
×
10^(6
−
3)
1 ×
10
6
is
bigger:
its
power
of
ten
is
bigger.
t
=
0.5 ×
10
3
0.5
is
under
1:
0.5 ×
10
3
=
5 ×
10
2
t
=
5 ×
10
2
Step 2 · Find times as big (T)
Use T = t, or 1 ÷ t when t < 1 (times as big = first ÷ second, or 1 ÷ first ÷ second when first ÷ second < 1)
The
first
is
the
bigger:
it
is
t
times
the
second.
T
=
5 ×
10
2
Check
Put the numbers back in. Both sides should match.
(1
×
10
6
)
÷
(2
×
10
3
)
=
5 ×
10
2
✓
5 ×
10
2
=
5 ×
10
2
✓
Topic check
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Related lessons
Scientific notation
Main lesson
Multiply in scientific notation
Use this for “(3.1 × 10⁴) × (2 × 10⁶)”.
Add and subtract in scientific notation
Use this for “3.6 × 10⁵ + 2.9 × 10⁵” or “How much larger is 4.2 × 10⁸ than 7.5 × 10⁷?”
Powers of ten by name
Use this for “Match 1,000,000 to one million and to a power of ten.”