Basic Multioperation Examples

Written By Sam Engel

Multioperation: Example 1

Consider the following question:

(48 × 4 × 61 × 70 × 61) / (30 × 66 × 23 × 65)

Trying to multiply the entire numerator, multiply the entire denominator, and then divide is inefficient. The goal is to reduce, rearrange, and “balance ratios” so the arithmetic becomes small and manageable.

Basic principles

  1. Cancel obvious common factors (especially 10s, 5s, 2s, 3s).

  2. Rearrange to pair numbers into clean ratios.

  3. Borrow by canceling near-equals (like 64 vs 65, or 65 vs 66), then correct at the end if needed.

Step 1: Obvious factors (10 and 5)

  • 70 on the left and 30 on the right reduce to 7 and 3.

  • It’s useful to remove the remaining 3 if possible: 48 / 3 = 16.

Now the expression becomes:

(16 × 4 × 61 × 7 × 61) / (66 × 23 × 65)

Step 2: Ratio balancing / borrowing

  • 16 × 4 = 64.

  • 64 is extremely close to both 65 and 66, so it can be paired with either denominator term:

Option (illustrative): pair 64 with 66, treating the ratio as roughly 64/66, and keep track of the correction if higher accuracy is required.

After the pairing/canceling idea is applied, the remaining “core” calculation is essentially:

(61 × 7 × 61) / (23 × 65)

Now calculate the main square:

  • 61² = (60 + 1)² = 3600 + 120 + 1 = 3721

So the remaining structure is:

(3721 × 7) / (23 × 65)

From here, the estimate approach in the original notes is:

  • Use a correction for the earlier “borrow” (the removed near-ratio), then approximate:

    • 67 × 23 = 1541

    • 3721 / 1541 ≈ 2.4

    • 2.4 × 7 ≈ 16.8

A further refinement observation (also in the original text):

  • Since 7 / 1541 is about 1/220, then 3721 / 220 is about 16.9

Main takeaway: the problem becomes manageable only after (1) stripping out 10/5 structure, (2) canceling the 3 cleanly, (3) pairing near-equals, and (4) computing the central square 61² quickly.

Multioperation: Example 2

(99 × 90 × 57 × 12 × 23) / (75 × 26 × 11 × 70)

Step 1: Obvious factors + simple factors

  • 99/11 reduces to 9/1

  • 90/70 reduces to 9/7

So the expression becomes:

(9 × 9 × 57 × 12 × 23) / (75 × 26 × 7)

At this stage, it can be worth doing one more simplification because it cleans up the remaining arithmetic:

  • Pull a 3 out of 75 (75 = 3 × 25)

  • Pull a 2 out of 26 (26 = 2 × 13)

  • Use those factors to reduce the 12 in the numerator

That yields:

(9 × 9 × 57 × 2 × 23) / (25 × 13 × 7)

Step 2: Ratio balancing / borrowing

Now use “friendly operations”:

  • Dividing by 25 is multiplying by 4 and shifting the decimal two places.

  • 57 × 2 = 114
    114 × 4 = 456
    Shift two places → 4.56

So the expression is effectively:

(9² × 4.56 × 23) / (7 × 13)

Compute the remaining ratio:

  • 9² = 81

  • 7 × 13 = 91

  • 81/91 is close to 8/9

So:

(8/9) × 4.56 is between 4.05 and 4.06

Then multiply by 23:

23 × 4.05 ≈ 93.2

(Which matches the note that the correct answer is about 93.3.)

Main takeaway: after reductions, the entire problem becomes “one clean division by 25” plus “one near-1 ratio” plus a final multiplication.

Multioperation: Example 3

(26 × 5 × 96 × 18 × 33) / (32 × 86 × 19 × 91)

Step 1: Cancel shared structure (“13s”)

Notice 26 = 13 × 2 and 91 = 13 × 7. Cancel the 13:

(2 × 5 × 96 × 18 × 33) / (32 × 86 × 19 × 7)

Step 2: Save a near-1 ratio for later

The terms (33, 18) in the numerator and (32, 19) in the denominator are chosen because their product ratio is close to 1.

So separate the expression into two parts:

Part A:
(2 × 5 × 96) / (86 × 7)

Part B:
(33 × 18) / (32 × 19)

Compute Part A:

  • 2 × 5 × 96 = 960

  • 86 × 7 = 602

  • 960/602 ≈ 1.6
    (and if needed, it can be corrected by multiplying by 150/151)

Now estimate Part B:

  • The original notes describe rewriting it as “1 minus a small fraction,” specifically:

    • (33 × 18) / (32 × 19) = 1 − (14/608)

Then adjust for the fact that Part A was approximated by 960/602 instead of 960/600:

  • That correction is reflected as shifting from 14/608 to about 14/610.

Finally:

  • 610/14 is a little more than 40

  • 1.6/40 ≈ 0.04 (slightly less)

  • 1.6 − 0.04 ≈ 1.56, and the answer should be slightly less than 1.56.

Algebra note (same content, cleaned up)

This is the conceptual identity being used when terms are in “difference of squares” form:

  • (a − b)(a + b) = a² − b²

  • If both numerator and denominator have that structure, the ratio can often be rewritten as:

1 − (b² − c²) / (a² − c²)

and then factored again:

1 − (b − c)(b + c) / (a − c)(a + c)

Main takeaway: when a product ratio is “almost 1,” rewriting it as “1 minus a small correction” can turn a messy division into a small adjustment.

Sam Engel
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Fractions