Library / /205 of 241

, 2020

Algebra Difficulty 2.7 Junior Find the answer Canada

How many points $(x,y)$ , with $x$ and $y$ both integers, are on the line with equation $y = 4x + 3$ and inside the region bounded by $x=25$ , $x=75$ , $y=120$ , and $y=250$ ?

$44$
$36$
$40$
$32$
$48$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

To determine the number of points $(x,y)$ on the line with equation $y = 4x+3$ that lie inside this region, we determine the number of integers $x$ with $25 \leq x \leq 75$ that have the property that $y=4x+3$ is an integer between 120 and 250.

In other words, we determine the number of integers $x$ with $25 \leq x \leq 75$ for which $120 \leq 4x+3 \leq 250$ and is an integer.

We note that, as $x$ increases, the value of the expression $4x+3$ increases.

Also, when $x=29$ , we get $4x+3 = 119$ , and when $x=30$ , we get $4x+3=123$ .

Further, when $x = 61$ , we get $4x+3 = 247$ , and when $x=62$ , we get $4x+3=251$ .

Therefore, $4x+3$ is between 120 and 250 exactly when $30 \leq x \leq 61$ .

There are $61-30+1=32$ such values of $x$ , and so there are 32 points that satisfy the given conditions.

Want a route through all this instead of an archive? The track puts 2,444 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.