Comparing improper fractions and mixed numbers

randRange( 1, 5 ) WHOLE randRange( 1, 9 ) randRange( M_NUM + 1, 10 ) M_DENOM/ getGCD( M_NUM, M_DENOM ) M_NUM/ getGCD( M_NUM, M_DENOM ) randRange( 1, 9 ) randRange( M_NUM_2 + 1, 10 ) M_NUM_2 / getGCD( M_NUM_2, M_DENOM_2 ) M_DENOM_2 / getGCD( M_NUM_2, M_DENOM_2 ) WHOLE_2 * M_REDUCED_DENOM + M_REDUCED_NUM M_REDUCED_DENOM getLCM( M_DENOM_REDUCED, I_DENOM ) LCM / M_DENOM_REDUCED LCM / I_DENOM F1 === 1 ? "remains as" : "becomes" F2 === 1 ? "remains as" : "becomes" M_DENOM_REDUCED*WHOLE+M_NUM_REDUCED (function() { if ( (M_AS_I*F1) > (I_NUM*F2) ) { return ">"; } else { return "<"; } })() M_DENOM > I_DENOM ? "more" : "fewer" M_DENOM > I_DENOM ? "smaller" : "bigger" M_DENOM > I_DENOM ? "less" : "more" SMALLER === "smaller" ? "bigger" : "smaller"

Fill in the blank.

WHOLE\ fraction( M_NUM, M_DENOM, false, true ) ____fraction( I_NUM, I_DENOM, false, true )

SOLUTION

First, let's convert the mixed number to an improper fraction with the same denominator.

To get the numerator of the improper fraction, multiply the denominator (M_DENOM_REDUCED) by the whole number (WHOLE) and add the numerator (M_NUM_REDUCED).

M_DENOM_REDUCED * WHOLE+M_NUM_REDUCED = M_AS_I

We can write the mixed number as an improper fraction with numerator M_AS_I and denominator M_DENOM_REDUCED.

Now we need to compare fraction ( M_AS_I, M_DENOM_REDUCED, false, true ) to fraction ( I_NUM, I_DENOM, false, true ).

It is easier to compare these two fractions when they have the same denominator.

Their smallest common denominator is the LCM of M_DENOM_REDUCED and I_DENOM.

\lcm(M_DENOM_REDUCED, I_DENOM) = LCM

The first fraction BECOMES_1 \dfrac{M_AS_I * F1}{LCM}.

The second fraction BECOMES_2 \dfrac{I_NUM * F2}{LCM}.

We see that \dfrac{M_AS_I * F1}{LCM} SOLUTION \dfrac{I_NUM * F2}{LCM}.

randRange( 1, 5 ) randRange( 1, 5 ) randRange( 1, 9 ) randRange( M_NUM + 1, 10 ) M_DENOM/ getGCD( M_NUM, M_DENOM ) M_NUM/ getGCD( M_NUM, M_DENOM ) randRange( 1, 9 ) randRange( M_NUM_2 + 1, 10 ) M_NUM_2 / getGCD( M_NUM_2, M_DENOM_2 ) M_DENOM_2 / getGCD( M_NUM_2, M_DENOM_2 ) WHOLE_2 * M_REDUCED_DENOM + M_REDUCED_NUM M_REDUCED_DENOM getLCM( M_DENOM_REDUCED, I_DENOM ) LCM / M_DENOM_REDUCED LCM / I_DENOM F1 === 1 ? "remains as" : "becomes" F2 === 1 ? "remains as" : "becomes" M_DENOM_REDUCED*WHOLE+M_NUM_REDUCED (function() { if ( (M_AS_I*F1) > (I_NUM*F2) ) { return ">"; } else { return "<"; } })() M_DENOM > I_DENOM ? "more" : "fewer" M_DENOM > I_DENOM ? "smaller" : "bigger" M_DENOM > I_DENOM ? "less" : "more" SMALLER === "smaller" ? "bigger" : "smaller"