Unknowns in × and ÷ Sentences, Both Sides Busy

Think of a number sentence as a balance. Nine times box equals one hundred and eight: nine boxes on the left balance one hundred and eight on the right. To find one box, divide both sides by nine, and one box is twelve. Both sides — that is the rule that keeps the balance true. The box was multiplied by nine, so you undo with division. If it had been divided by four, you would multiply both sides by four. If twenty-eight had been added, you would subtract twenty-eight from both sides. Free the box by undoing what was done to it, on both sides at once.

When one side is busy, make it a single number before you do anything else. Sixty-seven minus triangle equals thirty-two plus seventeen: the right side is forty-nine. Now the sentence says taking triangle from sixty-seven leaves forty-nine, so triangle is the gap between them, eighteen. Forty-eight divided by box equals four plus two is forty-eight divided by box equals six, so box is eight. The trap is to stop reading at the equals sign. And notice the direction: when the box is being subtracted or divided, it is the gap — sixty-seven minus forty-nine, forty-eight divided by six — which is different from the box being added or multiplied.

Some sentences do two things to the box. Three times box plus four equals twenty-five: the box was tripled, then four was added. Undo in reverse order — the last thing first. Subtract four from both sides: three times box is twenty-one. Divide both sides by three: box is seven. Check by running forwards: three sevens are twenty-one, plus four is twenty-five. If you had divided by three first you would be dividing twenty-five by three and getting a fraction, which is the signal that the order is wrong. This is the working-backwards lesson again, written as a balance.

Tests link two sentences with a shared symbol. Nine times box equals one hundred and eight, and triangle plus box equals thirty-six; find triangle. Start with the sentence that has only one unknown: box is one hundred and eight divided by nine, twelve. Now that box is known, the second sentence has only one unknown too: triangle plus twelve is thirty-six, so triangle is twenty-four. Always begin with the sentence you can solve on its own, and carry its answer into the other. Then check which symbol the question actually asked for.

Hardest of all is a box on both sides, and the balance handles that too. Two boxes plus nine equals one box plus twenty. Take one box off each pan, and it is one box plus nine equals twenty, so the box is eleven. Check both sides: two elevens plus nine is thirty-one, and eleven plus twenty is thirty-one. Three triangles equal ninety-six minus twelve: the right side is eighty-four, so each triangle is twenty-eight. Whatever you remove from one pan, remove from the other, and the sentence stays true. Quiz and exercises now.