Reasoning · Grade 6-1 Fraction and Decimal Calculations
Problem
Simplify messy fraction-decimal expressions
Six expressions mix fractions and decimals: 6.45÷2.55; (4.69×2.32+53.1×0.232)÷2.32; 3/4×2.84÷3 3/5÷(1 1/2×1.42)×1 4/5; (9999×1/6+3333×1/2-6666×1/9)÷7/9-3300; 1993÷1993 1993/1995; 99999.9÷5+9999.9÷5+…+9.9÷5. Each hides a repeated factor. Find the value of each.
Your answer
Watch outNo repeat seems visible until you shift a decimal point.
How to solve
StrategyOrganize Information in More Ways — Every one of these six is a trap for anyone who starts calculating immediately: the numbers are ugly on purpose. So before touching a pencil the move is to re-sort each expression until its structure shows — change a decimal into a fraction, move a decimal point from one factor to another so a repeated number appears, pull a common factor out of a sum, or split a number into a round number minus a small piece. That re-organising is tool 15, and what tells me which rewrite to try is tool 5, spotting a number that has already appeared somewhere else in the same line (2.32 in (2), 1.42 in (3), 3333 in (4), 1993 in (5), a whole row of nines in (6)). Once the structure is out in the open, each expression falls into two or three small independent pieces that can be handled one at a time, which is tool 7.
1STEP 1
Decide to reshape first and calculate second
Reshape before calculating.
Order of operations tells you what an expression means, but it does not force you to evaluate in that order; any rewrite that keeps the same meaning is allowed, and choosing a friendly one is what makes the arithmetic small.
Long division by a decimal is fiddly, but the same two digits sit under both numbers here, so writing them as fractions makes the hundredths cancel and turns the problem into reducing one ordinary fraction.
6.NS.A.1Organize Information In More Ways
3STEP 3
(2) Make the repeated factor visible, then pull it out
Multiplying one factor by 10 and the other by 1/10 leaves a product alone — that trick lets you make two products share a factor, and a shared factor can be lifted out of a sum and then cancelled against the divider.
6.EE.A.3Look For A Pattern
Making the repeated factor visible lets it be pulled out of the whole sum in one move.
Why?
So many of this plus so many of that is that many of the two together, so a shared factor lifts out of a sum.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
Why?
The factors of each term may be written in any order, so the shared one can always be moved to the front where it is easy to see.
🧱Commutativity of multiplicationa rows of b and b rows of a count the same array of dots — rows and columns just swap.
4STEP 4
(3) Re-pair each multiplier with a matching divider
Dividing by a bracket means dividing by everything inside it, so the bracket can be broken open into two separate divisions — and once every step is a multiply-or-divide, you may deal with the friendly pairs first instead of going left to right.
Repeated digits like 9999, 3333 and 6666 are a hint that one of them is the building block of the others, and pulling that common piece out turns three ugly multiplications into one easy fraction sum.
A mixed number whose whole part and whose numerator are the same number is really that number times something slightly bigger than 1, so the huge number is a common factor that vanishes rather than a number you ever have to divide by.
6.NS.A.1Organize Information In More Ways
7STEP 7
(6) Add first, using a round number minus a little
Numbers made of nines are one step below a round number, so trading each of them for "a power of ten minus 0.1" turns a column of awkward decimals into one easy sum and one easy division.
6.NS.B.3Look For A Pattern
Answer
2 9/17, 10, 1/2, 33, 1995/1996, 22221.9
3333 − 3300 = 33
Every answer passes a rough size check. (1) 6.45 is a bit more than twice 2.55 (which is 5.10), so a quotient of 2 9/17 ≈ 2.53 is right. (2) 53.1 × 0.232 is about 12.3 and 4.69 × 2.32 is about 10.9; their sum 23.2 divided by 2.32 is 10 exactly. (3) the chain multiplies by numbers close to 1 and divides by a 3.6, so a small answer like 1/2 is expected, and 1/2 is between 0 and 1 as the mixture of doubling and halving suggests. (4) the bracket is about 1667 + 1667 - 741 ≈ 2593, and 2593 ÷ 7/9 ≈ 3333, so subtracting 3300 leaves a small two-digit number. (5) dividing 1993 by something a hair bigger than 1993 must give an answer just under 1, and 1995/1996 is just under 1. (6) the five numerators total a little under 111110, and a fifth of that is a little under 22222.
Takeaway
When the same number shows up twice in one line, rewrite the expression so it can cancel — the ugly arithmetic was never meant to be done.
Decide to reshape first and calculate second
(1) Turn both decimals into hundredths
(2) Make the repeated factor visible, then pull it out
(3) Re-pair each multiplier with a matching divider
(4) Factor out 3333, then let the sevenths cancel
(5) Split the mixed number so the 1993s cancel
(6) Add first, using a round number minus a little